What if we swap “gender” for “cool”? I think it’s pretty inarguable that’s a social construct. I think I’m cool, and while walking around in socks and sandals isn’t cool, I know I’m cool nonetheless.

Yes, gender is inherently associated with sex, and correlates with it the majority of the time, but it’s not defined by it. This is similar to driving and being an adult - most adults drive, and most drivers are adults, but some grow up on farms, driving as kids, others live in live in accessible cities and never get their license.