Sticking a bunch of tokens in the school doesn't even strike me as out of character for the setting; wizards are so racist that they would absolutely prioritize random kids with even a spark of magic from a shit-covered village over their own neighbors. The books made it clear that wizards could not give less of a fuck about non-wizards, I doubt the average Hogwarts student could tell you the first thing about Scotland despite living in it. Rowling always cast it in a "hee hee how quirky!" way, but if you actually sit down and think about it for a bit, the wizards' willful ignorance of everyone else demonstrates an utter contempt. The ones calling everyone else mudbloods were just saying the quiet part out loud.
A troon wizard on the other hand makes no sense, given that they could just polymorph into whatever form they feel like. A failure to pass would represent that the person didn't want to pass, ie clearly a fetish for being a weird tranny who people know is a weird tranny. If dude wanted to be a woman, he could've been a perfectly convincing one.