I think it has to be more complicated than gender as a social construct. I know plenty of women who are very "male-like" in what they do, how they dress and how they look. But they still identify as being a woman. Mostly they are lesbian. They self identify as a lesbian woman, they do not view themselves as a heterosexual man stuck in an anatomically woman's body.
I've never had a conversation with a trans person about what it all means, though I do kind of know a few. But from what little reading on the subject I have done I do not believe it's as simplistic as feeling like "I fit better with the opposite gender norms of society therefore I must be the opposite gender." I think there's a deeper and more fundamental psychology to it, and probably even as we break those gender norms and accept a much broader range of expression by males and females we'll still have people who deeply feel like they are in the wrong body.
“The fundamental cause of the trouble is that in the modern world the stupid are cocksure while the intelligent are full of doubt.” - Bertrand Russell
"When the power of love overcomes the love of power, the world will know peace."
Jimi Hendrix