Would anyone class themselves as transgender if there were no gender stereotypes?

I work with a three year old who is, biologically, a boy. However, he always seems ashamed and genuinely embarrassed by the fact that he is male and makes every attempt to be like a girl (in the sense that he follows the stereotypes of what young girls 'should' do). His embarrassment clearly shows that it causes him a lot of distress.

I was wondering if people think that anyone would feel like they were born as the 'wrong' gender if these stereotypes didn't exist and the only thing defining gender was a person's genes. Do people think that they are transgender because they act in a way which society tells them is wrong, or is it a lot deeper than that?

((I'm not suggesting that there is anything wrong with being transgender, I'm purely interested to know if the problems that people, including young children, face regarding their gender is because of society's expectations or more than that.))

There is a difference between gender roles and gender identity. Moreover, the existence of intersex people proves that gender is much much more than biological sex.