Do you think sex didn't exist before 1912? It seems like you are confusing sex with the rough approximation of the shape of a chromosome and oversimplifying how those chromosomes relates to phenotype.
Trans people change their sex, not their gender. Trans men are men before they start taking T and trans women are women before they start taking E. What changes is their sex. Personally, I hope we see transgender replaced with something soon. Transsexual is fine as a subcategory of transgender people (or a category with lots of overlap with transgender people).
Social constructs don't all originate in the same way. If you want an example more like sex, you can look at the social construct of race. Race is not skin color, but the social construct is related to things like skin color. Just because race is a social construct doesn't mean skin color came after race.