What we do know is that individuals are affected by both the chromosomes and hormones of their natal sex and that these components do to some degree contribute towards establishing gender behavior. For example if you give testosterone to female mice, scientists have found that this increased their levels of aggression which is more associated with male behavior.
To say that gender identity is entirely a social construct would be a false statement as much as saying that it’s entirely determined by biology would also be.
In the 1960’s, John Money theorized that gender identity was not just about natal sex and could be heavily impacted by social conditioning. As witnessed by the case of David Reimer, who completely rejected his rearing as a girl upon being gender reassigned on Money’s recommendation, it is clearly about both.
Money defined the notion of gender role thusly:
"all those things that a person says or does to disclose himself or herself as having the status of boy or man, girl or woman, respectively. It includes, but is not restricted to sexuality in the sense of eroticism. Gender role is appraised in relation to the following: general mannerisms, deportment and demeanor; play preferences and recreational interests; spontaneous topics of talk in unprompted conversation and casual comment; content of dreams, daydreams and fantasies; replies to oblique inquiries and projective tests; evidence of erotic practices, and, finally, the person's own replies to direct inquiry."
That recent Israeli study I referenced was not conclusive but if anything it might have helped reinforce the idea that disembodied brains are not easily gendered. However this does not mean that there aren’t portions of the brain or elements at the level of DNA that help determine intrinsic gender traits.
It’s probably most realistic to think of gender identity as being influenced by both biological predisposition and social factors such as parenting style as well as sibling and peer interaction. Why some people have a propensity to reject their natal sex is a question we cannot answer. Just as we also cannot determine how much that rejection is tied to biological versus other factors.
One thing we might want to agree on is that expectation of gender behavior need not be coupled to natal sex which can go a long way towards alleviating the anxiety of gender non-conforming children. By removing objection to their behaviour early might we also be averting an eventual transition down the road or does that even matter?