> By that definition, isn’t “emo” — or literally any other social category — a gender

No, “emo” is a social category that is not exclusive with genders, its in a different bucket.

But, yes, the distinction of social categories into groups like “gender”, “race”, etc., is, like the categories themselves, fundamentally an arbitrary social construct.

> Do you believe that segregated services — sports, bathrooms, locker rooms, etc — were intended to be segregated by gender, as opposed to sex?

Binary “sex” is just ascribed gender on the basis of a subset of sex traits. To the extent there is a valid basis for segregating services, it varies from service to service. Similarly, the motivations vary from service to service (and, generally differ from the legitimate justifications, if any.)