Gender is entirely a societal construct. Many of the cultures that exist currently, or once existed and left artifacts in forms that we can understand them from a modern lens, have often modeled their gendered structures around biology, which I think is the point you are trying to make. However, basically everyone who has taken the time to study biology, society, and/or gender have realized that there is nothing inherently biological about gender.
Fun fact: the singular "they" is several centuries older than the more modern imperative towards a generic "he/him" (Shakespeare uses the singular they, for instance), and it has never universally been considered a-grammatical by the typical authorities on the subject, and in fact most of the style guides that have recommended avoiding it in the past have started to acknowledge it once again as perfectly acceptable in both informal and formal writing. At worst it's been described as clunky or sometimes unclear (which is rather the point, particularly in the case of the modern resurrection of the singular "they").