Why 10 Tishrei

We took בַּחֹדֶשׁ הַזֶּה as a hint to the dates of Pesach and Chag HaMatzot. The dates range from 11 Nissan to 23 Nissan, depending on how Shabbat falls. This is according to J. The same would, mutatio mudandis, apply to Sukkot, which would have dates 11 Tishrei to 23 Tishrei. Later, P (following D) fixed the dates to 15 Tishrei till 23 Tishrei, including an eighth day, and including an Isru Chag, according to P, who wanted to be maximally cognizant of the Shita of J. Yom Kippur is a day of introspection before the period of Sukkot starts, i.e. before 11 Tishrei, according to J. That is, perhaps, why Yom Kippur is 10 Tishrei.