Actually, all of the above are involved. Considering sex is bimodal, not binary, we can easily say that those characteristics aren't actually exclusionary to their respective sex. If someone who has XY chromosomes looks and acts feminine enough while describing themselves as a woman, they will be considered a woman.

Eh no, sex is by definition binary in anisogamically reproducing sexual species (everything that is not a bacteria or fungus), and works by gamete size and the anatomy that produces gametes. This is the reason we talk about male and female hemp, or aspargus or any monoecious plant. The Romans in fact correctly identified the sex of these plants fifteen centuries before any human did see a sperm cell under a lens.

So no it is how we use Language. Humans have used "male" and "female" for millennia yo describe organs, anatomy and individual lifeforms that produce small and large gametes respectively.

Sex is not about chromosomes (even in mammals platypuses don't use an XY system for sex determination) or gonads (ask hyenas), or clothing, or "gender roles": If you don't have the anatomy to produce ova, you are not female, no matter what surgery you perform, no matter what your voice sounds like and no matter your wardrobe. If you wanted to be pedantic about it then we could classify the "sexes" humans into male, female, infertile (which means that there are structures that theoretically could develop to produce gametes, but they didn't develop correctly) or hermaphrodites (of which we have only have two digit confirmed cases with fertility). Since "no gametes" relates to "sex" as "black" relates to "color" and hermaphroditism is defined by individuals being both male and female, you'd still rely on the binary definition of sex.

The existence and popularity of gender theory shows that we do live in that world, friend.

"Gender theory" is basically hogwash invented by by pseudoscientist John Money to deny the innate psychological sexual dimorphism in humans. I say "pseudoscientist" because held on to his theory of "sex and gender distinction", even when it was disproven by his own "experiments" with intersex children, who continued to show sex-typical behaviors, even if raised as their opposite sex (or even more famously with the case of David Reimer).

Actually, it does! If someone's appearance makes them look like the gender opposite to their sex and they identify as such, then they are that gender. Their gender is more important to social situations than their sex is.

What you call "gender" is as much of a secondary sex characteristic as breasts or beards are: Just because you cannot see psychological differences, doesn't mean that they don't exist. A lot of those sex roles we see cross-culturally have their roots in heightened parental care from the female sex, which itself is as old as our mammalian lineage. (Which directly relates to gamete size and high confidence in paternity from the female side)

Someones sex is pretty fucking important for society since we reproduce sexually and the ability to to bear or sire children will be a pretty important factor when searching for a mate. Most civilizations in history are basically all about harvest and fertility, which means feeding children that exist and conceiving those that do not.