Your seem to be confused between what you call a 'sex binary' and ideas around gender roles and streotypes. You seem to think that being male or female must necessarily result in certain types of behaviour or preference and thus is socially constructed.
Male and female are biological significators not personality indicators. Sex is indeed binary. What is not binary is personal character, preference or expression. This is the 'gender' bit.
Nobody here is promoting fixed sex based roles or expressions - in fact that seems to be what you are doing. Personal expression is indeed fluid regardless of sex. Some men like fashion and home interiors and dressing up, and some women are highly rational and logical and excel at design engineering, as two very stereotypical examples. Some men are same sex attracted and want to be desired by other men, and some women are same sex attracted and are romantically fascinated by other women.
But that doesn't over-ride the facts of sex which at a deeper biological level does have an impact and consequences. Being in a sexed body does determine certain types of experience - not only in a physical sense, but following on from that in a mental, emotional and social sense too.
We are not entirely separate from our body and its cellular level of programming; and modifying the hair style, the clothing or even the secondary sexual characteristics does not change one's sex at a deeper cellular level.
Sex matters, and it matters in quite specific ways for women and girls - which is why we have single sex protections and rights. society has deemed that women and girls are in need of these rights if they are to be afforded dignity in our society.