Anyone's a 'legitimate asylum seeker' if they discard all ID and cross borders illegally. You can't easily prove that they're not fleeing legitimate war or persecution.
Of course, fleeing the US in that way, or any way, over trans issues would be utterly idiotic, when the country is so large and diverse, and relocating within the US would surely be a better option, if they legitimately feel stuck in a high-crime or high-intolerance area.
Yes. Literally anyone can seek asylum in the first safe country they arrive in. Getting it is another story, even if you have a legitimate claim.
Almost like it's a privilege.