Yes, they're distressed because of this mismatch.
> But they are no more “evidence” than someone saying “god is love” is evidence that “god” is “love”, or that the speaker has any coherent notion at all of what those words could possibly mean.
You seem to be making an appeal to transcendental meaning - evidence isn't based on transcendental meaning or equivalence, it's based on observation of material reality. "gender" is a term created to describe the fact that trans people clearly have a mismatch between their sex and an internal property. That's all there is to it.