i think it's disingenuous to try and argue that 100% of the negative aspects of dysphoria are from outside sources. a big part of it comes from the reality that the person looking back at you in the mirror is not the person you want it to be, and that's entirely self-perpetuating. the actual medical argument that justifies medical treatment is to prevent depression, which is seen as otherwise unavoidable. one of the primary causes of the dysphoria is the identity mismatch. people that don't have that dysphoria can likely live happy lives as occasional crossdressers, meaning you're dealing with a different issue; the transgendered condition is explicitly defined by the dysphoria caused by the mismatch in perceived and actual identity, which to me sounds an awful lot like a mental disorder.
if there's not a disorder underlying it, it becomes a mere cosmetic procedure, and "gender dysphoria" becomes just another kind of teen angst. i think it's imperative that "i'm in the wrong body" is not understood as the same thing as "this dress makes me look fat".
obviously, it doesn't follow that there's a "cure", or at least not on the terms presented by the right. logically speaking, there are two answers: you can either try and stamp out the mismatched identity, or you can facilitate transition. the general perception is that the latter is usually more successful, and generally preferable, in an ethical sense. but, it doesn't negate the possibility of the first option - although it would require pretty heavy tactics, leaning towards an implementation of the brainwashing scenes in a clockwork orange or some other horrific shock tactics.
but, on a semantic level, the argument presented here is accurate. playing with gender is not a mental illness. rather, gender dysphoria is a type of depression with transition as it's only known (and ethical) solution.