I think a huge problem here is you seem to be blaming trans people for objecting to a weaponized characterization of mental illness, rather than the people who are literally, and intentionally, misusing and devaluing the term.
To use your analogy with being called gay, using it (and faggot and similar terms) as an insult, generic or not, is indeed marginalizaing toward gay people. But if someone who isn't gay objects to being called that, they're not the ones at fault. It's the ones using gay as a synonym for something bad. Similarly, trans people being called mentally ill by people who are using the term to devalue them and invalidate their experiences are not at fault when they object.
Put the blame on the people who are actively placing stigma on mental illness by using it as a synonym for "what you feel isn't real and you're not valid", where it belongs.