You're just repeating the same assertion though. As I said, this doesn't match my understanding of gender critical feminism at all. I'm not saying you're wrong, you may well know more than me about this than me, but I'm asking you to flesh out your reasoning.
Why do you say they base their definition of womanhood only on sex-based oppression as opposed to them recognising that women's sex-based rights are based on sex-based oppression? After all, if there were no sex-based oppression--and here, we're not just talking about sexism, but men's greater ability to attack and rape women--there'd be no need for sex-based rights.
As for whether it’s sex based or gender based oppression, it depends what you’re talking about. Female sports for example, are entirely sex-based. Because of their sex, the top 1% of females aren’t as strong or as fast as the top 1% of males. Gender identity has nothing to do with it. Female sports exist so that females can get the spotlight and opportunities they deserve as sports people.
A space like a bathroom, is gender-based. As I've argued before, I think that trans women should be able to use the bathroom that conforms to their gender identity. As many of them already do. Because bathrooms (with cubicles) are mostly private spaces anyway. As long as you look "feminine enough" nobody will challenge you in a female bathroom. And, indeed, some more butch women do get challenged in female bathrooms.
But it gets a little more complex with spaces like changing rooms. Here, women and girls undress in communal spaces. Women have, for centuries, had the perfectly reasonable expectation that when they are vulnerable and exposed in this way, they will be allowed to do so away from the view of men. Even if a man has no intention of taking advantage of the situation. It's the same reason why considerate men will cross the street or walk on ahead if they find themselves walking behind a woman on a dark, secluded street. It's not because we're a danger, it's because we care about her comfort.
Again, here I say men as distinct from trans women. I'm not denying the existence or experience of trans women. I'm not saying that trans women are perverts who are faking it to gain access to women's spaces. I'm saying that greater care than, "anybody who says they're a woman can access female-only spaces," is needed in order to keep women safe and protect their right to comfort.
And no, I don't think that Rowling fails to recognise trans women's need for safe spaces at all. No decent person does. And Rowling has stated explicitly, numerous times and perfectly clearly, that she believes trans women are vulnerable in much the same ways that women are, and that they need and deserve protection. It's a question of balancing these intersecting needs. Trans activists seem utterly uninterested in doing so.