I think that the scholarship on display here is more than adequate to the task of explaining why the term "Tudor" does not appear in the writings of the 16th Century. Clearly, one would have to have a fairly significant amount of time spent with the writings and the materials of the day.
However, the written word does not equal the spoken word. A term like "Tudor" could certainly have been spoken or used when two people spoke of the monarchy, but the formal, written words used during that time may have avoided the term. We know that people cursed; rarely do you see written proof of what they said informally. We know they had sex; rarely do you find examples where the details are laid out in the text. We know there were jokes, songs, and cutting remarks made, because that is part of human nature. Funny how they were not written down all that much.
The official records and writings of the time may not reference "Tudor" but perhaps some of the more cutting remarks made between rivals included the term. Perhaps some of the songs of the era used the word as a shorthand or a mocking term. Saying conclusively that the term was not used is a little hard for me to believe, but I do accept what Mr. Davies is saying.