Are you just distinguishing something that is partly socially constructed with something that is completely socially constructed? I have a very hard time following any of your complaints, and I can't figure out whether the bottom line is that you agree with Carmelo's definition of gender but think it is fair to characterize that definition as "socially constructed", or if you don't agree with the definition.
I also can't tell if you read Carmelo's post. For instance, you say, "The real world is complex, and reducing every human behaviour purely to biology denies the agency of humans" as if it's a contrast to something Carmelo says. But Carmelo says, "humans are complex, conscious, social animals. They are highly susceptible to social conditioning". Um...same...thing? So why do you make it sound like he's missing something?
You say, "To reduce behaviours or traits purely to biology and place biology above the social in terms of the importance of human development and civilisation is both incredibly simplistic and incredibly dangerous." But where did Carmelo do anything remotely like that? He doesn't even mention civilization!
And you say, "You use similarly biologically reductionist arguments to imply that certain gender identities are ‘natural’"...but where did he do that, either? Do you mean the stuff about testosterone that was bolstered by a couple of links to evidence? That you didn't refute at all? And wasn't linked to a claim of a natural gender identity?
And isn't your point "The assertion that some gendered behaviours are actually biologically innate - an assertion you yourself make - is a regression to this outdated understanding, one which denies the complex interaction between the biological and the social." actually just a restatement of Carmelo's point: "Rooting gender in societal norms requires us to ignore the reality of biologically rooted gender norms that may appear to be socially constructed, or may overlap with social norms in a way that is difficult to parse out." (Emphasis mine.) He points out a complex interaction between the purely biological and the social...so...how is he denying it?!?! He says rooted not caused, indicating that something else is going on (which, in context, must be social); he also also, with "overlap" acknowledges that multiple influences contribute to behavior.
Did you even read his article, or just skim it and decided to attack what you guessed it might say?
If you did read it, and you have substantive criticisms, maybe you can try again and actually directly address his points rather deliver high level characterizations that seem really off-base.