Intellectual coherency on transgender issues will be rare because so much of it is agenda-driven on many sides, even conflicting within the same side. In a way it's starting to resemble the ferocious debate on climate change. Classifying it one way or the other has grave implications for the loser(s) and it becomes easier to "stretch" your claim when the spectre of that is raised. It's still important to keep in mind that the phenomena underlying the debate is objectively demonstrable: sex-based differentiation in the brain has been observed in ordinary men versus men who transition to women. It's also important to keep in mind that a simple philosophical argument highlights the inconsistency in saying that a man who wants a certain body is diseased for wanting the same things that would cause distress if they were absent in a woman.
It remains unseen if redefining womanhood will be a sustainable solution for trans' individuals integration and equality, but the fact that it is considered is reassuring that we have the right priorities as a culture oriented toward the individual, toward secularism, and toward liberty. However, we must be willing to admit what is working and what isn't working. Aspects of transition, such as hormone therapy, clearly work. Other aspects of transition, such as ignoring sex differentiation to the disadvantage of others in sports, may not be working so well. Honesty combined with good-faith and individual, secular values will be important here.
(Disclaimer: I am transgender and I consider myself to be more in a third-category best described as "different from most men")
Whatever adjudication process that exists in a culture that doesn’t care about date of birth would probably reach a different judgement.
Yes, extremely rare ones. 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.
Gender expression is a different, more complex matter. There has always been "grey" areas from an anatomy perspective and I think there is consensus that other factors affect this as well. Race, depending on context, is even more complex. Is it about nationality? Geography? Skin color? Religion?
Most things about people don't fit in neat boxes. Age is not one of those things.