The OP doesn't discuss how sex is assigned at birth, so I fail to see how that's relevant; it's a tiny facet of the much larger issues that it does discuss.
Lots of things are complex, but that doesn't mean that they're social constructs. Quantum gravity is a good example; so is biological development, which results from the interaction of genetic and environmental variables (this is why "Nature vs. Nurture" is a meaningless dichotomy). Biological sex results from such interactions; this is simply a fact. Is it messy, and not dichotomous? Of course it is -- there are all sorts of variations that result from that developmental process. However, it's very much a bimodal distribution: the vast majority of the members of the vast majority of species fall into two categories: those having female reproductive organs and female secondary sex characteristics, and those having male ones.
Sexual reproduction depends on the existence of two sexes; given that reproduction is, umm, critical for the survival of a species and of the DNA of individuals, it makes sense that evolution would have selected for a clear perception of "male vs. female" by its members. But over the past century or two, as agriculture has ceased to be the way most people make a living, at least in the developed world, having lots of babies has become detrimental to the success of a family (as well as to the survival of the species).
So reproduction is devalued in modern societies. Because of this, there's now a lot more "wiggle-room" in our notions of "male" and "female," and it's becoming possible for people to accept that those notions are rooted in more than biological determinism. This has recently led to the differentiation of gender, which is culturally based, from sex, which is the result of a developmental process that's biological in nature. The concept of gender is a useful one: it acknowledges that much of what goes into the perception of male vs. female is culturally determined (or "socially constructed," if you like). To argue that biological sex is socially constructed is to advocate throwing out the concept of gender: the latter ceases to have any meaning if we choose to believe that there's no such thing as biological sex, and that it's entirely a matter of arbitrary agreements among members of a given society. I think this would be a major step backward in any serious effort to understand and discuss how people experience "femaleness" and "maleness."
The OP is a classic example of an argument that purports to be factually based but is entirely driven by ideology.
(Note that for humans and other social animals, the social world forms a significant part of the environment in which individuals develop; it certainly influences brain development, and it may well be that there are ways it influences sexual differentiation -- that's actually a very interesting question, but it's an empirical one.)
