thatdamnwoman (OP): Is there anyone out there capable of boiling down both his argument and her response in plain English so that I can intervene with some sanity on FB?
I will try. First you need to understand the author of the piece is a (self-described -- see the 'Comments' to the piece) 'social constructionist'.
[Social Constructionism (or, similarly but well, in some ways perhaps different; never mind Social Constructivism) is a philosophical dogma which holds that reality is a social construct. (No, really, these people do think that or at least they say they do. Whether the claim that reality is socially constructed is actually a claim, or amounts to just something like saying ' Jabberwocky Pinstripe ', is controversial. Let that go for now.) However, reality is not a social construct. Hold on to that thought. The earth really does go round the sun on a roughly elliptical path -- that it does has nothing to do with our concepts, our language, or our social mores. Humans are really sexually dimorphic. And so on. Reality is not socially constructed. OK, now. Relax.]
Next step in his argument. There is a logical rule sometimes expressed as ex falso quodlibet. Roughly what this means is, if you start with a determinedly false (set of) premise(s), anything you like follows logically. (I used to start explaining this to students by saying something like, 'If the moon is made of green cheese, then I am Icky the Fire Bobby '. That is a valid deduction the conclusion follows from the premise but only because the premise is false. There's more. But that is a start.)
So, now, the argument in question is based on (nonsensical or false) premises which are not true (premises which together make up, or exemplify, the daft social constructionist creed). The conclusions follow from these premises, but only because the latter are not true. This does not mean the conclusions are false, notice. The author has given us no reason for thinking them true, however. And, in fact, most of his conclusions like many of his premises are not even false: they are just nonsense.
Stock's response generally makes a laudable attempt to draw the debate back to earth and deny that reality is socially constructed. You can perhaps do the same on FB, OP. Just remind people who are impressed by this rubbish that (1) the whole argument depends on the (risible) notion that reality is a social construct; (2) the conclusions mostly make no sense, even if validly derived via ex falso quodlibet.
One more thing. The referenced article is the sort of crap that gives philosophy a bad name. If you want to see what some of the grown-ups in academic philosophy think of this nonsense, check out Felix Leiter's blog here. See, for instance, this, which Leiter describes as 'funny and apt'.
[Leiter's blog covers much more than the trans issue in philosophy. It is more-or-less the go-to blog for (anglophone and other) professional philosophers: he is good for getting a view of what grown-up philosophers think of the 'woke', though.]