I think the traditional belief is that, the first time, Moses wrote it all himself either in stages over the course of 40 years or all at once while up on the mountain. (With the exception of certain verses Chazal point out that were not written by Moshe ie those about his own death - although there is mahloketh over this question).
The question about the manner in which Moshe wrote the text leads to an interesting discussion about the origin of the Oral Torah which the Chazon Ish touches upon (by his standards he was touching upon it and left it sort of open ended, but he shows tremendous depth and make very thought-provoking observations probably no one else could think of). I wish I could give over a coherent synopsis, but we went through this with my rabbi on Shavuot night and I couldn't take notes, and it was extremely complicated!
The way you describe your resolution seems to me not really to resolve the problem. Supposing that multiple scribes were transcribing Moshe's lectures and they inserted their own biases, interpretation, etc. we would still be left with the incorrect notion of multiple and contradicting authors of the one text. If you say that they merely transcribed his speeches word for word without any alteration or any insertion of personal opinion, interpretation, bias, etc. then that is no different than if Moshe had written it himself. It's not a problem of handwriting which leads to this speculation, it's seeming differences in different parts of the text. So I'm not sure I totally follow your idea. In any case, it is one of the 13 ikkarim of emunah listed by the Rambam that the Torah was written by Moshe and not altered. And I don't believe we have sufficient reason to doubt that.