Of course. And there is no reason not to assume that the kunyomi original "te" has the whole limb as its reference field. But that does not address his question -- which flows from the written characters -- not the purely oral language.
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. You are going about this in exactly the wrong way. First the words existed, and then characters were assigned to them, in a very ad hoc manner. The 手 character does not wholly represent the Japanese concept (nor the Chinese, I daresay), it is merely an orthographic marker for that concept.