No, the characters are irrelevant here; they are merely orthography. The Japanese words for body parts preceded any Japanese written language, native or Chinese. "Te" is not a "kunyomi" unless we are talking about how to vocalize characters; the word existed before the 手 character was used to write it, and exists now beyond the character.
I am not arguing that -- and I agree with you that "te
" is not a Chinese vocalization. But his question
had to do with the disconnect in the orthography
-- 小手 -- from a modern text, and not the preexisting kunyomi
or the "pure nihongo," if you prefer, (whatever it was). It is not a Chinese compound, and at the dates you are talking about -- it could not have been, as they were not used. That makes it a synthetic Japanese expression -- in Chinese script.
We have very little way to know the "pure nihongo" that does NOT go through the manyogana
for the preliterate early Japanese. Struggling with the orthography is unavoidable in etymology of such basic words. Kojiki-Den
almost certainly has the word "手" discussed -- and I wonder what Norinaga has to say about it?