Sarah I don't think my use of 'gender identity' and 'gender' are the same as yours? Could be wrong. They are two different things to me.
I think you can have a gender identity of female but be expressing a quite masculine gender. Eg a 'butch lesbian' that has short hair, no makeup, wears mens clothes sometimes (I have met some that do), and not conform to any stereotypes of 'female gender role' at all but they still have a female gender identity.
So yes I agree that a large part of gender is social construction but that's not the same as core gender identity.
I've heard some professionals describe it as an onion. the outer layer is the most superficial aspects of one's gender identity, makeup, hair style, clothes etc.
The next layer in is your social role and how you want to be related to on a day to day basis. Next layer in is secondary sex characteristics like needing to have, or surgically remove breasts. After that the inner layer is primary sex characteristics, what genitalia feel right for you. I can't remember if that is the core layer or if the core layer is inside that one - 'core gender identity'.
Not sure if I remembered that quite right but that was the general gist. It might be seen as out of date now but I think it's a useful model. Whatever our identity is in terms of that onion model I also think gender identity development is a type of imprinting and once you have imprinted your gender identity it is generally permanent (that's my theory equivalent to yours, which I am also fairly convinced is fact!).
I also believe (there is some evidence) that the development of that gender identity is influenced by genetics, biology, the reality of one's sexed body, prenatal hormones, environment and culture. Some people can develop an identity in conflict with that sexed body.
I think it's the two outermost layers in that onion that 'gender' lies that you describe - gender expression, stereotypes about appearance and gender roles. Historically linked to the inner layers of course - but feminism seeks to change that.
I agree with your last paragraph - having gender non conforming role models and for all of us to be gender critical so we are not shackled by societal norms/ patriarchy is a good thing.