As you say, they are "the five books of Moses". I regard it as completely irrelevent whether he personally penned them, or had a scribe write things as he said them. The point is, they are direct revelation from Hashem, as given to "his servant, Moshe".
More than that is simply making "a tempest in a teapot".
I would even be quite content to discover (if it were true) that most of what is in the Torah was initiated orally, and only written down later. That is quite compatible with the facts we have, as well. It would still be "the books of Moshe", containing, "the Word of Hashem, as given to Moshe".
Every week in the synagogue liturgy we proclaim,
V'zot ha torah,
Asher sam Moshe lifnei B'nai Yisrael
Al-pi Adonai, b-yad Moshe
This is the Torah,
Which Moses placed before the children of Israel,
From the mouth of the Lord by the hand of Moses.
It has never changed.
The fact that the transmission occurred is more important than exactly how it happened, I think.