It's fine to identify as your sex, but I don't think it proves the GC's stance that gender identity does not exist

You may not think it, but I believe the GC people believe it. The GC position, as I understand it, is that there is no way for a person to know their sex other than by their anatomy, because there are no non-anatomical differences between the sexes. The idea that there is "maleness" or "femaleness" (or any other "-ness) runs counter to the GC view that women are oppressed because of their anatomy, and, but for that oppression, there would be no difference between men and women. I find that notion absurd, but I believe that is the GC notion.