I watched the video. It's easy enough to see where it goes wrong, I think. At some point the bloke with the big hair and the moustache says, "Everyone has a gender identity."
This is false. Not because some people lack a gender identity, but because there just is no such thing as gender identity, at least in the sense of a kind of feeling humans have that may or not match somehow with an individual's sex.
(The case is actually stronger, if you think about it: there could be no such thing, since feelings and sexes are different categories (as Gilbert Ryle might have put it). But 'there is no such thing' will do for our purposes.)
I know some people believe in gender identity. We have to deal with that, sure, just as we have to deal with people who believe in guardian angels, ghosts, fairies, devils, gods, prophets, ... whatever. We have learned to live with people who believe in guardian angels -- not by accepting their beliefs as cogent, but more along the lines of, "That's nice dear, now run along and play," sort of thing, barring them from teaching their beliefs to our children, and so on.
So for other false beliefs. We correct our children, allow space in our societal institutions for believers - however daft - and get on with things in as reasonable a manner as we can.
In this case, "That's all very well, but, you see, there's actually no such thing as gender identity in the sense you're thinking about," seems a more than adequate response to all the (consequential) nonsense.
No? -- Leave it at that? Why not?