Feminism and Diversity

From a "radical Feminist" viewpoint. I don't mean this to be offensive and believe that anyone should have the right be who they are.
From a bigger society institutional point of view, does the societal gender dichotomy cause many transgendered people to feel that they have to be transgendered? What I mean, for example, is that a female child might hate dolls and dresses and prefer sports and the outdoors and be less passive than most of her female peers. People around her tell her that she is acting like a boy and should stop acting like that and be a girl. If that is what being a girl means, she does not want to be a girl As an adult, she chooses to transition into a man instead. If those around her instead told her that there are many different ways to be a woman, would that result in a life long female gender identity instead of male.
Do narrow societal standards of gender make more people transgendered than looser standards? Do "masculine" women and "feminine" men that remain their birth genders loosen societal gender standards while transitioning accepts narrow standards?
I have seen the comments about some Feminists being transphobic. While I am accepting of anyone being who they want to be, I admit that I have this issue. I don't want to be transphobic. Please educate me.