What's the value of learning "Chitas" compared to any other daily Torah study?
Side question: If it's so important, why was it introduced only recently?
R' Zev Dov Slonim compiled a booklet that collates various sources that talk about the advantage of learning Chumash, saying Tehillim, and Learning Tanya daily. The booklet is also included in the Chitas sefer, published by Kehot. The footnotes on Sefer Haminhagim's entry on Chitas discusses this as well.
Here are some of them, see sources inside for more:
The Shibolei HaLeket (Inyan Tefilla #44), brings from Rashi that the early Chassidim would learn for an hour after prayer. He connects this to the idea brought in the Talmud (Kiddushin 30A) , that one should always split his his learning into three. A third Mikrah, a third Mishna, and a third Talmud.
Rabbi Slonim points out that in Shulchan Aruch HaRav (Hilchot Talmud Torah, Chapter 2, Halacha 1), the Baal HaTanya says that explanations of the Written Torah fall under the heading of "Mishna" and Kabbalah falls under the catagory of "Talmud".
Hence, learning Chitas daily covers Mikrah (Tehillim, Chumash), Mishna (Rashi on Chumash), and Talmud (Tanya).
With regards to Chumash, from here:
"Studying a parshah of Chumash every day with the commentary of Rashi, reading Tehillim every day, and completing the entire Book of Tehillim on Shabbos Mevarchim, - about these things one must be scrupulous. They are crucial to oneself, one's children, and one's children's children." (HaYom Yom, 25th of Shevat)
"The Alter Rebbe once said: 'The commentary of Rashi on the Torah is the very wine of the Torah; it unlocks one's heart and reveals one's innate and elemental love and awe [of G-d]. The commentary of Rashi on the Gemara unlocks the mind and reveals one's innate and elemental intellect.' " (HaYom Yom, 29th of Shevat)
"Studying a parshah of Chumash every day with the commentary of Rashi activates the light in the soul and the revelation of the soul - which is a glow of the revelation of Mashiach." (Kovetz Lubavitch, Booklet 7, p. 24.)
On Tehillim (see here for who split it up into a daily portion and why):
From the Chida (Midbar Kedemot, Entry for Taharot (Tet), and Tehillim (Taf)) David Hamelech prayed and requested that saying Tehillim should be considered like learning Negaim and Ohalot (Taharot). The students of the Arizal said that learning Taharot is a Tikun for spiritually blemishing the Brit. David Hamelech was asking that his Tehillim should also peform the same spiritual healing.
the Chida then quotes Rabeinu Efraim on Parshat V'Zot HaBracha, who says that one who recited Tehillim daily is considered as if he fulfilled the whole Torah. See there for the connection.
the Chida follows that with a quote from Rabeinu Efraim on Parshat Vayigash, who says that one who recites Tehillim daily will merit to be under the Kisei HaKavod (G-d's throne). See there for the connection.
ויהי חתת אלקים על הערים - בראשית לה:ה
On the above Pasuk, R' Zevin relates a story titled "The Secret Weapon" in "Sipurei Chassidim al hatorah" as follows:
Late in 1856, Reb Menachem Mendel of Lubavitch (known as the Tzemach Tzedek) sent his son Reb Shmuel - who was later to succeed him as rebbe - to S. Petersburg on matters of communal concern.
On the eve of his departure he told him the following: "when I was summoned in 1843 to the rabbinical conference in S Petersburg. I went to pray at the resting place of my saintly mother at Liozna, and she told me that by virtue of her self-sacrifice for chassidim and for Chassidism she had been privileged to be granted entry to the palace of the Baal Shem Tov in heaven. She had requested the Baal Shem Tov to pray on my behalf, and had asked him to give her some sgulah that would enable me to withstand the opposition of the antagonists of Chassidism. The Baal Shem Tov had answered: 'Your son knows - by heart, and letter perfect - the Five books of the Chumash, all the psalms in the books of Tehillim, and the book of Tanya. Now we read in the Torah, that the sons of the Patriarch Yaakov were not harmed by the Canaanites because of 'the dread (חתת) of G-d was upon the cities around them.' The tree consonants of this word are the initial letters of the Hebrew names of these three books: Chumash, Tehillim, Tanya. He who is thoroughly acquainted with every letter in them is granted the spiritual strength to shatter all the obstacles that prevent the revelation of holiness."
"Therefore," Reb Menachem Mendel instructed his son Reb Shmuel, "wherever you may be, whether in government institutions or with ministers of state, recite a chapter of Chumash, a psalm from the Book of Tehillim, and a chapter of Tanya."
Many years later Reb Shmuel recounted this episode to his son, Reb Shalom Ber, and added: "It was a wonderful prescription. With the first three chapters of the Chumash, three psalms from the Book of Tehillim, and three chapters of Tanya, all the plans of the maskilim collapsed, and the Children of Israel, true to the Torah, came out triumphant."
Learning Chitas is something that is in the realm of every Jew. Before the revelation of Chassidus, "learning" was limited to the Yeshiva Elite and the average Jew was not able to learn Torah. Why was it only until recent that such a formula was established? This I do not know but Chitas in one way or another was done by Chassidim before the Rebbe Rayatz (6th Lubavitcher Rebbe) set up an official system.
Chumash with Rashi has been learning for hundreds of years and is a topic of discussion in the Shulchan Aruch as to if Chumash with Rashi fulfills the Torah requirement of "shnayim mikra v’echad targum" Learning the text of the weekly parsha twice with the targum.
The simplest Jews have been saying Tehillim for hundreds if not thousands of years. One of the reasons being that while there are opinions that a person must know the words of Torah he learns, this is not the case with Tehillim as simply reciting the Hebrew has a power on its own which is why 75% or more of our davening is from tehillim.
Tanya while a new Sefer is nothing new as the Alter Rebbe says in the very beginning all he did was compile sources into a small book.
What is the value? If you find value in learning Torah then there is a great amount of value. If you do not ascribe to the Chassidic value of learning that is available to all of Klal Yisroel then your question should be reworded to whether learning Chitas is a good use of your time instead of learning Gemara with Tosafos and Poskim.
As a Chasid, I would say for any Jew it has immense value.
It's more than just learning. It's also a connection to the Rebbe.
This perhaps can answer both your questions.
I would like to address both Chitas and outreach:
To study the Parsha daily 1 aliya per day -with Rashi was what many Gedolim did,
Tehillim, was something we did whenever hardships were upon us.
Tanya was studied daily by Chabad Chassidim.
Came the 6th Lubavitcher Rebbe and based on a lashon of the Tzemach Tzedek 3rd Rebbe that said - "Vayehi Chitas hoelikm" - "The fright of Hashem was on the nations" - Chumosh Tehillim Tanya is Roshei Teivos Chitas-" so the Frierdiker Rebbe made the takana to actually learn Chumosh based on day of prasha's aliya - day.
Tehillim broken up in 30 cycle of the month,
Tanya yearly cycle (a cyacle for a normal year a cycle for leap year) broken down by the Rebbe.
He felt it was important due to the Matzav- situation, Russian persecution, Hitler etc. that more protection is needed.
In the next post - outreach
A Tzadik a Nosi b'Yisroel who knows down here what these generations needs are, and he knows the highways of heaven, establishes a study - Hashave lechol nefesh- that everyone at whatever level they are, can, and should learn, (in addition to regular shiurim) it becomes somewhat a minhag yisrael, at least to the people who heed his call.
In addition, it brings the blessings of there shall not be "Chitas" from our enemies and the blessings of the Tzaddik who gave many times a bracha to all who accepts the study.
You can run through the study azoi vi a vasser - just reciting the words- in fact thats what happens with Tehillim- and you can do the same with the chumash and Tanya, however the proper way is to underatand and delve in deeper.