Women who insist that they specifically get to decide who is or isn't a woman and what women believe aren't new. Phyllis Schlafly managed to ensure the Equal Rights Amendment didn't pass on this same basis. Phyllis would fly from city to city, addressing crowds of women to tell them that women should be at home looking after their kids, not um, flying from city to city making political addresses like she did...
Beware anyone who claims to represent "all" of some large diverse group, such as "Women" or "Floridians".
"Women should have single sex spaces" turns out to be used to justify, "It's OK to be hateful and even violent against women in these spaces so long as your excuse is that you believe they're not actually women" which is bullshit.
Years ago, when I wasn't too tired to spend all day and half the night dancing, I went to Bang Face Weekender - basically imagine a huge multi-room club night except for days and days. I keep the socials for it available because hey, it's a nice memory. This sort of "Single sex spaces" bullshit caused a problem for the last-but-one Bang Face because a new-to-this Security outfit somehow decided it's their job to go remove people who in their view weren't women from a toilet for women. These women weren't causing any problems for anybody else, but because they presumably had the wrong genitals or for some other reason were "suspect" to that Security team, Security dragged them out of a toilet cubicle and threw them out of the site. Other clubbers were of course horrified, and the event runners had to apologise to everybody - because regardless of how many X chromosomes you have, or whether you do or don't have a womb, dragging people out of the toilets because you've got weird ideas about what is or isn't a woman is batshit.
Phyllis Schlafly is an odd comparison to make. She argued that women should stay in traditional roles and out of public life (while as you mention, not following her own advice), whereas JKR and other feminists take the exact opposite view. Not sure I see the relevance of your analogy here.
As for Bang Face last year, what happened is that security staff kicked a group of males out from the women's toilets. I agree that this isn't an ideal outcome, much better would have been if these men had respected that women's spaces are not for them, and stayed out in the first place. The fact that their removal was treated as some sort of scandal shows how far we've lost sight of the rights of women and girls to have single-sex provisions.