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Many (all?) Ashkenazi communities have a different and distinctive tune used for keriat hatorah on Rosh Hashanah and Yom Kippur morning (and in some communities also on Hoshana Rabbah and Simchat Torah).
However, I’ve observed that this tune is not used at mincha of Yom Kippur. Why not?
I've heard two answers from a respectable rabbi, zt"l. Even after death, I'd rather keep his name and location anonymous.
One answer I heard is that the High Holiday tune has a "mode" of "royalty" and "joy". This type of tune is inappropriate when reading about illicit marriages that people should avoid. (I will add in, B"N, as to why this section is read for afternoon of Yom Kippur, later.) As I don't completely understand this mood of "joy" in the tune, I have trouble appreciating or understanding this answer, so, perhaps, accept it (or not) for what it is.
The second answer makes a bit more sense to me. We have to understand the whole reason as to why Yom Kippur afternoon is getting a Torah reading in the first place as well as what was read. According to this article reading the Torah on Yom Kippur afternoon was a much later addition first mentioned in a Braita and later in Bavli Talmud Megila 31a.
We know that the Torah is not read on Mincha of any other Yom Tov. So, this leads to the question - is the reading of Torah on Yom Kippur in imitation of Shabbat or a fast day? Read the rest of the article which cites mainly a debate by Ismar Ellenbogen. Regardless, it is clear that the reading during the afternoon is not particular for the occasion of Yom Kippur. Thus, the tune used is not the Yom Kippur tune but the same tune used for Shabbat and fast days.
Personally, I feel that the reading is more in line with it being a fast day. Support for this is thinking, perhaps, is in the idea that the brachot for the haftarah end in Magen David which is more indicative of a fast day. However, even the article debates the origin and need of any haftarah in the afternoon on Yom Kippur at all.
Anotehr possible idea is in viewing the article, it was common to finish the reading of Devarim on Yom Kippur prior to the advent of Simchat Torah. While I see that you mentioned that some havethe custom of using the High Holiday tune on Simchat Torah, I have found that this is a "less" common minhag. At least, I have seen this done in a minority of roughly 60 various Nusahc Ashkenaz shuls that I have attended throughout about 50 years. I think that one of the reasons places are doing it is to somehow hold on to the "flavor" of the previous month's melodies. However, I don't think this Yamim Nora'im tune is appropriate for Simchat Torah as the spirit of the day has nothing to do with the concepts of awe and royalty, assuming that's the main reason for the special tune, in the first place.
