There is the slight problem of Deuteronomy 34:6 and the record of the death and burial of Moses. So you know at least that much wasn’t written by Moses.
And in answer to your original question, I believe Moses was the authoritative source for the Torah but not necessarily the author of the whole written text. Since he was the source there is no problem with Jesus referring to Moses and the Torah in the authoritative sense. I don’t believe Jesus ever said directly that Moses actually wrote the Torah. In that culture the source is what is important not the scribe that wrote it all down. To you perhaps knowing that a scribe wrote the text would make a difference.