I think the issue with this discussion is that "trans" is used as both a description and identity and is sometimes used interchangeably. But it's not quite the same thing. Similar to the word "gay". You can be gay whether or not you like being identified as such. If you're into someone of the same sex to the exclusion of the opposite sex, you're gay. However you feel about gay culture, the gay identity, whatever -- the description fits and thus you're still gay.
Trans is a similar thing. If you have a persistent, non-superficial desire to be a different gender than the one you're living as, you are trans. You don't have to identify as trans for it to the case; in fact, some of the most vehement transphobes I'd ever met were in fact themselves in the closet. Classic gay pastor syndrome I guess. To be perfectly honest: I've met some really toxic/crazy people in the trans community, often enough to be sympathetic to someone who might reflexively react badly to the label. I can only apologize for their bad experience, though most of us are just folks trying to live our lives. But you can still be trans while disliking the community or even hating the word "trans". Because, well, see above. It's a description as well as an identity.
One reason I really like this manga's premise is that this curse would be great at helping people figure themselves out. If it works the way I think: If you aren't trans but have some gender feels, you'll turn back as soon as your desire is fulfilled. If you are trans, it's permanent -- for the simple reason that your desire to express feminine is endless. (keep in mind that a whole village disappeared because nobody ever turned back. I wonder if their main export were eggs?)