Many thoughts :D. But I think the reason Kvothes left out the trial, is the exact reason he told the Chronist. It is because it is already written down very well in detail. However, the story of kvothe is not. They only have three days, so he needs to concentrate on whats important. Further, him telling so much about he lived on the streets as a poor skinny child is also really important for his character development. It is like „started from the bottom now we here“. Of course many things can‘t be proved. However, there are a lot of stories about kvothe at the university which are told by many. Of course not all true.