I interned for a year with a public speaker who had five or six autobiographical stories he told regularly. I found it interesting that his stories ended up like how the gospels tell Jesus stories. His messages had a main point, but he spoke extemporaneously, so depending on what he had been saying before he got to the story, he would included different elements into the story. One of the last messages I heard, he incorporated a few pieces of context at the beginning that I had never heard him tell before, and although I had heard him tell that story multiple times by that point, it completely changed the meaning of the story. Not that it invalidated the previous tellings, but that bit of context made a big difference to the meaning of the story. The gospel stories read a lot like that. Jesus may have told the stories differently depending on the context, and/ or the writer may have told the story different depending on the points he was making with the story.

Which is to say, I think it less likely that Q was written. Mark is generally said (by people who follow Christ, at least) to be summarized from Peter's messages. It seems likely that Matthew and Luke took from Mark as well as a shared source of apostolic teaching, especially since Luke claims to have researched these things, and at least several of the original disciples are traditionally said to have been preaching in the Greek-speaking areas of the Mediterranean.