Even if you don't want to call that gender for some odd reason, that doesn't change the fact that this social construct is separate from sex.
Now it’s “if you’re a girl and you like to play video games, maybe you were just born in the wrong body.”
Gender can’t be just a social construct and also a reason to transition, at least in my mind.
Like, i saw a meme on reddit with a person saying "People keep telling me to just be a butch lesbian instead of transitioning to a man :("
Presumably a butch lesbian can do anything a man can do except walking around without a shirt, so I don't think that's the issue.
There is some tension between the two positions, but they are reconciliable. As far as we have gender as a social construct, and we will for the foreseeable future, gender identity is inevitable.