Yeah, that's basically the extent of it. It's mental illness, but in the case of trans folks it's close enough to issues of sexual orientation that people argue for self-determination rather than against it, since "woman in male body" and vice versa are relatively compatible with extant gender politics (though note that this wasn't always or universally the case), while people whose fixation is on a self-image of them as an animal, as being physically different in a way not related to biological sex, &c. don't fit into any major social movements, so they get treated as mental illness instead.
Mechanically it's not really different from someone who thinks that they're a cat, Napoleon, or a different race, but it's an expression of dysphoria which is more or less harmless (since both men and women are ultimately human beans who interact with the world, society, and other individuals in largely the same ways). Once we get to the point where we can grow new mindless bodies wholesale there's basically zero reason for it to ever be an issue (and the same, ultimately, for a lot of other issues of dysphoria).
The main distinction, I think, is related to self-harm, harm done to others, and ability to function in society. Each of those cases Sonlirain mentioned involved the individual in question causing measurable harm to themselves, to others, or to their ability to live normal lives (or some combination of the three). Swapping your biological sex doesn't affect any of that, unless you live in a shithole where conservative social dogma will ruin you.
Hell, never mind trans people, if we can grow cat bodies capable of hosting human minds, communicating easily, and maybe possessing opposable thumbs, let someone with species dysphoria do as they please too.
One interesting point is the question of racial dysphoria. I seem to recall it being a thing, but you don't see support for it, because it doesn't align with the authoritarian false-progressive dogma--a trans-race individual would be derided and shamed worse in those circles than anywhere else.
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Basically what I'm saying is that the problem is that there's a stigma attached to "mental illness". A lot of types of dysphoria are more or less harmless, so there's no problem with the treatment essentially being the acceptance of the changed identity.