This process is already gatekept, to my knowledge. In my state, changing your legal gender id or name requires going through the courts and a substantial amount of legal paperwork - even if you do so without a lawyer, there are some substantial fees associated with that process. Not to mention if someone wanted to become less identifiable, they probably wouldn’t want to do that in a way that is available as public record. Personally, I’d probably just get a haircut - they’re a lot faster and a lot cheaper.
Additionally, banning everyone - especially banning exclusively trans people - from fixing their documentation is not a reasonable solution to your hypothetical problem, a fact so obvious anyone arguing in good faith almost certainly would have caught it.