There are three massive problems with a gender-is-social hypothesis being seriously entertained absent incredibly tight evidence.

First is that gender is universal among human societies. The roles change somewhat with the circumstances and history, but this looks a lot more like humans-have-language than like humans-use-subject-verb-object-order-in-language. Generally when you find something universal, it's not "just culture".

Second is that sex-specific behaviors are universal among mammals. Although it is not inconceivable that we have moved traits from being hardwired to socially driven, for the most part, we are good mammals. We think baby mammals are cute (not just baby humans), we are afraid of snakes, we like sweet things, and so on. Our evolutionary heritage runs richly in us, so the default hypothesis is that our sex-specific behaviors do also.

Third is a bit trickier, which is that a social implementation for something doesn't mean that it's not biological. For instance, many humans find the idea of sex with a sibling offputting. But it is societies that fashion this into a high-reliability norm that sculpts behavior into what is required by biological reality (i.e. that of inherited genetic diseases). Avoidance of procreation with siblings is, to a large extent "cultural", and yet it is equally as much "biological"...it's just implemented largely via culture. Critically, it is not arbitrary. Likewise with some grooming norms, or adherence to "the golden rule".

So I agree that the verdict is out, to some extent, on the degree to which gender expression is plastic and the degree to which gender expression impacts plasticity in the brain. But I don't think that the verdict being out means that we need to act tentatively regarding whether gender is purely cultural-and-arbitrary. Two strong lines of evidence indicate it is not, and one line suggests that even to the extent that it is cultural it isn't necessarily entirely arbitrary. For most things, that is enough to take a tentative stand. And thus it should be here, too.