[This post is my long-winded attempt to get to the truth by reference to online definitions, only I seem to have got to the same answer that a very cynical and simplistic analysis would have got me to].
Some example definitions of gender identity - "an individual's personal sense of having a particular gender."
"Gender identity is a way to describe how someone feels about their gender. For example, some people may identify as a boy or a girl, while others may find neither of these terms feel right for them, and identify as neither or somewhere in the middle."
"One's innermost concept of self as male, female, a blend of both or neither – how individuals perceive themselves and what they call themselves. One's gender identity can be the same or different from their sex assigned at birth."
And a definition of gender - "Gender is the range of characteristics pertaining to femininity and masculinity and is about social and cultural differences rather than biological ones."
So a gender identity is an individual's personal sense of having a particular set of characteristics that are nothing to do with the biological body, but do have something to do with society's expectations of different biological bodies or on a social or cultural level. Society's expectations based on someone's sexed body are by definition based on generalisations and stereotypes.
So, a gender identity is by definition an individuals personal sense of having a particular set of characteristics that accord with gender stereotypes (or do not accord with gender stereotypes in the case of the most special of all the trans people - the enbys).
If we forget about NBs and other neo-genders and pretend that all trans people are TM or TW, then a gender identity is by definition an individuals personal sense of having a particular set of characteristics that accord with gender stereotypes that are normally associated with the opposite sex.
Now, obviously, nothing in the real world should be based on "an individuals personal sense of..." anything, it should be based on truth and compassion and fairness and reason and things like that.
If we completely eliminate an individuals personal sense of how they accord with sex-based stereotypes, then we have to abandon gender identity as way of organising society. Which leaves the trans community with gender expression or nothing. Access to women's toilets comes down to whether you're wearing a dress or not?