> It's not clear to me what the existence of the occasional XXY individual, etc., has to do with the idea that gender is just a social construct.
It doesn't, but it does have to do (potentially) with the argument.
One side says "Gender is male/female, scientific fact, and can be examined in DNA". They are outright rejecting the idea of gender being a social construct.
The other side COULD try to argue that gender is a social construct by virtue of all the social variables and treatments that are involved, but that doesn't actually set up gender as a social construct - those can all be social issues ATTACHED to a non-social construct, just as young/old are the social constructs attached to chronological age. To convince the other side, you need to separate gender as a concept from the genes. If the "static binary" group cannot imagine a gender identity separate from the genes, they might understand someone XXY having one of those binary options. They may even understand one XXY person being "male" and another being "female". Even if they argue that there is still a rigorous "male" and "female" option that is based on the expression of the genes, they are accepting that we are making some definition of "male" and "female" that is from a collection of criteria. The "decision" of what is in those collections of criteria are more clearly socially influenced.
So: Society decides what defines "male" and "female" traits (based on "traditional" traits exhibited by chromosome-based sex) => society is defining male and female for the unusual genetic mixes => male/female is based on behavior => male/female isn't solely decided by chromosomes => having only strict male/female definitions is not the only set of definitions available.
At least, that's how I understand the argument. And I have some anecdotal evidence for it - My first exposure to transgender/intersex/etc was in college with a video made by a group of people who WERE unusual genotype or extreme phenotype differences. Listening to someone talking about how they felt like a "boy" but their penis was removed shortly after birth, forcing them to be in a world that treated them as a "girl" gave me a perspective I could identify with (and was horrifying - it also gave me my first real moment of empathy to consider how day-to-day life is different for a woman. Realizing that I (cis-straight male) thought the idea of being treated as female was actually scary caused me to rethink some assumptions...and I was raised to be socially liberal. The idea that there was a penis to be removed made me instinctively agree that this person could be male in a way that is hard to match.
That said, I don't think the non-XX/XY phenotype argument is the most persuasive available, or at least not for everyone. I do think it's a valid approach, even if such visible phenotype differences are rare.