if it’s good enough for the majority of historians
It isn't. Historians would love to have independent evidence of the existence and crucifixion of Jesus, but there isn't... so most historians refrain from taking a position one way or the other. The ones that do have to make do with what little objective information they have, and the best they can come up with is: well because of this embarassing thing, it's more likely that he did exist and was crucified than that he didn't, because why would they make that up?
That's rather weak evidence, and far from "proof".
Not sure why you’d need more
Well for one because the more prominent people who have studied this have a vested interest in wanting it to be true. For example, John P. Meier, who posited this criterion of embarassment that I outlined in my previous comment, isn't really a historian but a catholic priest, professor of theology (not history) and a writer of books on the subject.