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NOTE: This answer brings a commonly held opinion, and should be treated as a non-comprehensive, non-rigorous introduction. **It goes without saying that nobody should use this answer as a personal permission to go ahead and learn/teach Kabbalah - CLYOR**
One strong and common opinion is that yes it is permitted, some say obligatory, and certainly *strongly urged* (**certainly in contradistinction to the opinion in the question that it is strictly forbidden**. The minutia of the opinion in this answer can be debated and this answer will not deal with that). It is what will bring the Moshiach to our very low generation, who need "a very great light". However, there is a litany of cautions and guidelines and limitations, and one must follow them. Some of them are collected below. However, let's start with the sources:
## Sources ##
Collected from my own learning and verification where I have access, and the verification of others where not.
1. ***Or Neerav*** is a work by the Rabbi Moshe Cordovero (the Ramak) who taught Kabbalah to the Arizal. It's goal is to speak to, advise and encourage both teachers and students of Kabbalah. It writes extensively in Part 1 about the importance for every Jew to learn Kabbalah, and speaks out harshly against anyone who discourages it. He makes a distinction between easier parts and harder parts, but generally encourages everyone to learn the Kabbalah, even Maaseh Merkava and Maaseh Bereshit. He explains it is important to follow the Tikkunei Zohar's rules of learning Tanach and Mishna/Gemara, and the art of *pilpul* first, but emphasises one shouldn't use excuses like "I haven't mastered *shas* yet" etc. - a "portion" of these is fine. He explains that one shouldn't fear making a mistake, because he will eventually arrive at the truth, and the mistakes will be wiped out (he bases this on *Sefer ha-Bahir*, a very early Kabbalistic work). He does warn against not teaching it if one doesn't know what they are talking about, and exhorts students to find a teacher who has mastered Kabbalah for its own sake, and not as one subject among many.
2. ***Eitz Chaim***, the major Kabbalistic work of the teachings of the Arizal by his
foremost student, R' Chaim Vital T'zl, *Shaar HaHakdamot* where [he writes][1] that ideally one should learn Talmud first, and then move on to Kabbalah. Everyone should do so; anyone who doesn't causes sorrow to the Divine and delays the Redemption. He adds that if a person has issues with learning Talmud, they should go straight to the Kabbalah. He brings many sources for this, disagreeing with the "only after 40" idiom (see Shach, Yoreh De'ah 246:6).
2. ***Even Shleima*** by the Vilna Gaon T'zl, 8:24 and 11:3, as well as his commentary on Mishlei 4:22, 5:18, 7:12 and 20:9, and on *Tikunei Zohar* 107:3 (plus many other places). In his commentary on *parshat pikudei*, 17 and Mishlei 2:9 he writes that it is obligatory for everyone to learn Kabbalah1. His position amounts to the fact that the learning of Kabbalah will be the primary cause of the Redemption. In *Kol HaTor* he predicted that in the year 5750 (1990) the "revelation of Kabbalah" will begin.
3. ***Ohr HaChama*** by R' Avraham Azulai T'zl writes in the introduction that the decree of not openly engaging in Kabbalah was lifted in the year 5250 (1490)2, and in 5300 (1540) it became praiseworthy to do so in great numbers in order to bring Moshiach.
4. ***HaSulam*** by Baal HaSulam T'zl, where he writes in the introduction that the time has come for *everyone* to learn Kabbalah1.
5. ***Eder HaYaker v'Ikvei HaTzon***, p.144 R' Kook T'zl explains that it's no longer appropriate for the "big questions" to only be answered to the great Rabbonim, but everyone should have an answer, and all of klal Yisrael should participate in this process.3.
6. [The letter][2] of R' Yisrael Ba'al Shem Tov T'zl to his brother in-law, Rebbe Gershon of Kitov T'zl. In it he states that the Moshiach will come when the wellsprings of his teachings have spread [and been taught] to the *whole world*. The teachings, which he states he wasn't permitted to give in his own lifetime (and he didn't have writings as such, his teachings were transmitted through his students, who *taught* it to their Chassidim, and still do to this day), are Kabbalistic in nature.
7. ***Shaar HaEmuna VeYesod HaChasidut*** by Rabbi Gershon Chanoch Henoch (the Radzyner Rebbe), the son of the second Ishbitzer Rebbe and the grandson of the Mei Hashiloach writes in his introduction to the Beit Yaakov that it is imperative for everyone to learn Kabbalah and Chassidut, and like the Ramak above, argues harshly against those who discourage its study, and insist on focussing on *pshat* and *halacha*.
8. ***The Zohar*** itself, in several places explains how the teachings of Kabbalah will bring the Redemption. See [1:117a][3], taken to mean that in the year 5600 (1840) the wellsprings will burst forth from above and below, meaning Torah wisdom that was hidden will become available (Kabbalah, note the same word used in the Ba'al Shem Tov's letter), and this will coincide with a great influx of "lower" wisdom. See also 3:124b which states that specifically the teachings of the book of the Zohar (a primarily Kabbalistic work) will bring about the Redemption1!
9. R' Chaim of Volozhin T'zl, before he was niftar, told his son R' Yitzchak that it is extremely important that the entirety of his work ***Nefesh Hachaim*** be learned by everyone. It is a work that contains a lot of Kabbalah, which is written in such a way that clearly demands there be a teacher to explain it. I will look for that source again bli neder. His son made efforts to pass on his urgent message that this work be published and learned in full by all.
## Dangers ##
There are several noted dangers about teaching Kabbalah that cause us to hesitate and make sure we do so wisely. Here are just a few:
1. **It can be misunderstood**. Chacham Hillel *Shlita*, in *Faith and Folly*, goes into much detail about this danger. Two of the greater issues are:
- One might misunderstand the metaphors used and consider Hashem to be in some way composed of parts, *chas veshalom*.
- One might come to consider that Hashem has partners on His level, *chas veshalom*.
Kabbalah is a deep, mystical subject that deepens one's knowledge of Hashem, and misunderstanding it can lead to misunderstanding Hashem - Who wishes to be known - a great sin. Much of the original writings are written in a form of linguistic code, that requires a cipher to understand. If it is not learned with the explanations of the Tzaddikim that have come since then, then one has no hope of understanding anything, and is in great danger of *misunderstanding* something and falling into blasphemous thoughts, **chas veshalom**.
Do note on this that the Ramak says that it is actually part of the process; it is impossible to learn Kabbalah without making mistakes, so therefore one should still learn it, and Hashem will guarantee he will eventually succeed and his mistakes will be erased.
2. **It can become profaned**. These are very holy, personal, and even *private* (mature) teachings, about Hashem Himself, and should always be treated with extreme awe and trepidation. By exposing these teachings to "popular" culture, they can (and unfortunately have in many cases - "Madonna grade Kabbalah") become profaned.
3. **It can put people off**. See the story of the Splattered Gem by R' Pinchas of Koretz T'zl and explanations. This is a three fold issue.
- If it taught to someone not ready or through a faulty teaching or inexperienced teacher, it might put them off Torah (or worse, come to disdain it) rather than bring them closer, which is a huge tragedy and must be avoided.
- These teachings are considered some the highest teachings of all, and lay out openly the secrets inside the Torah. After hearing them, if a person continues to sin afterwards, he has no further excuses or mitigating factors of "he couldn't have known better".
- In addition to the last point, if a person is unmoved by these teachings, then it is considered a tremendous waste. If the deepest truths about Hashem do not move someone, then what will? Those teachings have been wasted, and that soul has little hope.
4. **Kabbalah itself is dangerous**. Kabbalah is a very powerful wisdom, and can be misused, and certainly should not be available to malevolent sinners who would do so. See the aforementioned *Faith and Folly* for much detail about this. It has also been the cause of several disasters in our history, such as the Shabtai Tzvi false messiah tragedy.
5. Another commonly cautioned issue is **it can confuse and distract**. Perhaps even make one go loopy (see the story of the *Pardes* in Chagiga 14b), or go off the derech. It can distract one's focus away from learning halacha, which is an absolutely critical part of a persons' service. This is something I have heard personally from one of my own Rabbonim, who is close with Chacham Hillel and other leading Kabbalists, and is also a strong point in Chareidi Litvish yeshivot, including my own. Just because this is not the only opinion on the matter does not mean the caution should be disregarded!
## Guidelines and Limitations ##
It goes without saying that one should not teach what one does not know very well. Therefore, before teaching *anything*, one should have discussed it properly with one's own Rabbis (who themselves obviously need to be experts in these topics), who should offer feedback, including as to whether one is actually good at explaining and teaching these things. There is a difference in being able to recite the *sefirot* and *olamot* and their connections, and actually understanding that and being able to translate it into a plain language every-day lesson. If one can't answer intelligent questions on a subject, they probably don't know it well enough to teach it - this is true of any subject.
1. **One should only teach to the level of one's audence**. As the Ramak states in Or Neerav, Kabbalah is a big subject, with some subjects that are easier, and some that are harder. There is much "easy"4 Kabbalah available nowadays, and there are many sefarim aimed at the public that contain a lot of Kabbalah, including works by the Maharal T'zl, the Ramchal T'zl , R' Chaim of Volozhin T'zl, the Chassidic Rebbeim and more. There are contemporary works by well respected Rabbonim such as R' Aryeh Kaplan T'zl, R' Yitzchak Ginsburg *Shlita*, R' Mattis Weinberg *Shlita*, R' Moshe Miller T'zl, R' Manis Friedman *Shlita*, R' Akiva Tatz *Shlita* and others. If one wishes to teach Kabbalah to the public, one should follow their guidance and use their explanations and not stray further without asking one's own Rav.
2. **The Ba'al Shem Tov has stated one shouldn't teach Kabbalah without his teachings** (which later became known as Chassidus). See *Derech Mitzvotecha, Shoresh Mitzvat haTefillah 2* by the Lubavitcher Rebbe.
3. **One should know one's audience**. Ideally, one shouldn't teach it to people who haven't got a good basis in Talmud and Halacha5, as well as trained in how to think and reason, as cautioned by the Ramchal. Where this is not possible, one should teach Kabbalah in a *very* distilled form, free of jargon, with very well thought out parables and plain, every-day explanations following after the great teachers of Kabbalah outlined in points 1 and 2, and make sure to include disclaimers about the dangers outlined above.
4. **One should become well acquainted with the dangers**, and their common causes. Now that all Kabbalistic works are available publicly online, and in popular books, there are a lot of questions and there is a lot of potential for misunderstanding, so it is important to know how to answer some of the main misconceptions, and offer some explanations to those who read these works and have questions, so they don't err.
5. As with all hashkafa, **one should always bring the lessons back to the practical**. Their main goal is to inspire and bring people genuinely closer to Hashem and thus to enhance people's adherence to halacha, the performance of Mitzvot, prayer and chesed.
**In general, don't do it alone**. Just like one shouldn't learn it alone, one shouldn't teach it alone, but with guidance from one's teachers, and without straying from the words of the Tzaddikim who have, themselves, written teachings of Kabbalah designed for the public.
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## Background ##
Based on the sources and my general understanding, the general traditional understanding of "don't learn Kaballah until you are a big Talmud chachamim post 40" has indeed changed, but how exactly and to what degree takes explanation. It should be noted that this wasn't always the case. It appears that at the time of R Shimon Bar Yochai, everyone did indeed learn Kabbalah, and it is a matter of exile that it ended up going "underground" (see Chapter 14 of Shaar HaEmunah VeYesod HaChasidut, introduction to Beit Yaakov). There was a bit of an awakening of Kabbalistic teachings in 12th century Spain, and at that point there was a pushback (by Yitzchak the Blind), and started a tradition of aversion to teaching Kabbalah to the public, and obscuring any recorded teachings behind omissions and cryptic language. This was in part because of the Mishneh in Chagiga that states that Kabbalah should only be taught from teacher to student orally.
The dangers section illustrates many of the main issues this tradition was established to protect us from. Given that those dangers still exist, it is still fair to say that even according to this opinion, there is still a lot of areas of Kabbalah that shouldn't be taught, and plenty of room for caution and hesitation. However there have been two major counterforces, as well as a good reason, that have worked to bring our cited opinions to make the changes.
Firstly, there has been a lot of clarification of the Kabbalah. As an esoteric subject, it's very hard to understand. However, after a few thousand years, the Tzaddikim have slowly been coming up with ways of explaining it that are better and safer. Also on this, lots of amazing and very useful tools for understanding Hashem, ourselves, relationships and more have been crafted based on the Kabbalah, and this certainly counts as "teaching Kabbalah", and certainly can be an argument towards "this kind of thing was always allowed", which is to say the "change" we are discussing here might not be as drastic or controversial as we might be saying. The Sefirot and psychology, the Olamot and relationships, come to mind.
Yet also, in these concepts are answers to many of the big questions about life, like what does Hashem's love really mean, and even what is the essence of the purpose of life that we are given in the *nigleh* answers to that question. This is important and is likely where most of the controversy of the "changes" come up, but clearly one can see why some opinions would be all for "taking a risk" and allowing more access to Kabbalah, so people can get answers to these important questions in times of need.
Secondly, there has been a lot of improvements in the general wisdom and intelligence of the world (see Zohar 1:117a). A lot of people think that mysticism and intellectualism don't work well together, however, the more intellectualism we have, the safer Kabbalah is (Ramak described Kabbalah as a "science" often), and the easier it is to understand, because actually, most of the difficulty in understanding isn't because it is _irrational_ (it's not, although it intersects with the super-rational), but because it is _abstract_.
Note: This counterforce actually leads to ***an obligation to teach Kabbalah***. This is because there has been a great increase in people learning Kabbalah, starting even several hundred years ago. The Nefesh HaChaim, who admits he is in the camp that is opposed to encouraging study of Kabbalah, claims that he feels that he is *obligated* to teach it now that many are learning it and getting it wrong. For example, in part 3, he explains (I am simplifying) that some people, upon learning Kabbalah and the omnipresence of Hashem, end up with erroneous opinions such as it is ok to contemplate holy matters in a dirty alleyway (surprisingly, this was a question I had in yeshiva! Anecdotal, but evidence that indeed our minds are more questioning, and more abstract these days).
So, due to the various intellectual revolutions in the secular world over the years, as well as improvements in education, literacy and access to knowledge, we've moved on a bit from the dark ages, can think way more abstractly, and now more people are capable of understanding more Kabbalah, and able to benefit from it, as well as not fall into the pitfalls of the past.
Which leads to the good reason: Kabbalah knowledge is important to know because it brings us closer to Hashem and redemption than anything else, as the sources go to length to explain.
Built in to this opinion is a lot of optimism, which is indeed an item of _personality_, and can lead to personality clashes. As someone in the middle I feel this very strong, but I must admit I am attracted to the notions that our hard work during the galus is leading to [tangible improvements][4] to the world, that people are becoming more good (even while they are becoming more numb and less naturally connected to *ruchnius*), and that there are answers to the big questions and more people can have access to them. Yet I can hear the other side too, it's important to never leave the ground even when looking upwards, and to never be irresponsible with the great dangers we've seen in history, and never belittle the warnings from our leaders past and present.
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## Conclusions ##
It is indeed the case that we have a strong historical tradition that claims teaching Kabbalah openly is a grave sin. However, we also have a strong tradition of permitting it, with explanations of why this has changed (or that this change was always intended, and to what extent it is a "change" etc), along with guidance, limitations and rules. We shouldn't ignore this, nor the fact that many great Tzaddikim for at least half a millennia have poured their hearts and souls into finding ways to find ways to explain essential and core lessons from Kabbalah so that the masses can understand them. With the Zohar itself claiming that our modern generation is nearly 2 centuries into an era of a great upsurge in holy and secular wisdom, which go hand in hand, and claiming that learning and teaching it is a critical aspect of bringing about the Redemption, we certainly should take this to heart.
However, as I opened the conclusion, this is indeed not the final say, and is certainly a highly charged issue, and there is strong disagreement on many aspects. Therefore as always with answers on this site, one shouldn't take this as their final say and charge, but should **speak with ones own Rabbi and find out what one's own tradition says on the matter, and follow it**, as well as taking into account all the answers to the question, not just this one. I am not writing as a "Rabbi" in authority, but as a student myself, of my own Rabbis who have their own traditions on this matter, which I strive to follow without error.
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1 - It is logical to assume, as many have, that once the public is allowed/obligated to learn it, the learned are allowed/obligated to teach it, as it is dangerous/impossible to learn Kabbalah by oneself, and it is generally bizarre to set up a system where people can learn something that nobody is allowed to teach. This answer does not deal with whether it should be taught actively or passively or in between - see Or Neerav and Sefer HaBahir for more treatment of this.
2 - It is not clear if this is overruling the Mishna in Chagiga or not, if the Mishna's ruling is absolute or not, or if the Gemara opens nuance to the Mishna's statements. Ramak for example seems to permit teaching these subjects in Or Neerav.
3 - See also *Igerot haRa'ayah*, vol. 1, Letter 43. He was a proponent of demystifying Kabbalah and introducing it to yeshiva curriculums.
4 - Easy means a) getting to the heart of a particular message in Kabbalah and b) finding a way to explain it without jargon, through mashal, in a way that the every day person can absorb it. The essence is very much preserved. The Tzaddikim have given us many examples of this, especially Chassidus, that we can borrow, and this late in the game, much has been distilled. There are different degrees to which things are distilled and much to debate about this, including what prohibitions still exist in which areas. *Faith and Folly* and the responsa of the Lubavitcher Rebbe in particular offer a range of information on this as a starting point.
5 - This has always been the case, the difference now being that everyone is now able to learn the Kabbalah, rather than a select few. Many have taken this to mean we should introduce Kabbalah to the curriculum, and even distil core lessons in Kabbalah for everyone. It is still *lechatchila* to have this grounding, and the more one has, the deeper one is capable of going into the Kabbalah. Debate can be had about how much Talmud and Halacha counts as "being well grounded".

[1]: https://www.sefaria.org/Sefer_Etz_Chaim.1.1.5
[2]: https://www.inner.org/baal-shem-tov/letter.php
[3]: https://www.sefaria.org/Zohar%2C_Vayera.30.445
[4]: https://judaism.stackexchange.com/questions/134338/aliyat-ha-dorot/134892#134892