I think Ada already said it better than I will, but yeah there’s a few too many simplifications and short-circuits (in reasoning/logic) in what you have written. If society stopped claiming people “should” be this or that just because of some physical characteristics they were born with, the “conflict” you outline ceases to exist. Trans people (and “non-conforming” people in general) could fulfill whatever roles and stereotypes they resonate with, and still fit in just fine.
Plus, the stereotypes some of us resonate with simply do not neatly map onto society’s stereotypes of men and women. Some people who have outwardly transitioned may seem to be moulding themselves into society’s “traditional” gender norms, sure. Putting aside those who do so out of self preservation or fear of persecution, I would argue it’s just happenstance (in other words, pure luck) that who they want to be in life lines up so neatly with these existing delineations. It doesn’t conflict with the push to stop expecting and demanding people fit other people’s schemas.
I don’t know that stereotypes are necessarily harmful, not do I think that the point of equality is to get rid of them. Rather, they should have no bearing on our worth, and we should not be reducing people to the various stereotypes we know of. We are infinitely more complex and varied beings than any stereotypes that anyone has ever conceived.
Myself, I prefer to think of stereotypes as archetypes, like in storytelling. They are shortcuts that evoke certain character traits. They cannot describe the totality of a person, only facets. Thus, we should be careful to not expect people to fit “inside” them; therein lies the/a root of a lot of harm. If that is too difficult, better to do away with thinking in terms of stereotypes than demean those who don’t fit them.
Thankfully the birds are often content to do most of the talking