You make some good points but I'm not convinced that you've fully grasped the concerns of JKR-minded people. I'm not sure I have either, but let's have a think about this.

Firstly, it's important to recognize just how astoundingly male-biased sexual violence is. Something like ninety-nine percent of perpetrators of sexual violence are male. Yes, you read that right. 99%. (Source: https://stoprape.humboldt.edu/statistics#1)

Given this, if you are concerned about being a victim of sexual violence, you are very very very much safer if you are around a woman than if you are around a man.

Furthermore, although most men aren't rapists, those who are will often engage in all kinds of other unacceptable behavior in order to commit the crime, including stalking, lying, drugging victims, breaking and entering, and so on. Although not every perpetrator is like this, there are certainly quite a number of manipulative, advantage-seeking people among this population.

So, here's the setup. Men are 100x more dangerous than women, on average. The men you actually have to worry about include a large number of manipulative ones who will do some prep work to enable their crime.

Okay, now, suppose we do the following.

Whoever says they're a woman is a woman. No medical checks necessary, police will agree that the person is a woman if they say so. That's it. Say it, maybe cosplay as a woman for a couple hours, and you gain access to every woman-specific space there is.

Remember those manipulative male rapists, the ones giving date-rape drugs and following their victims down deserted alleys? You think they'd never, ever, ever abuse this "say it and it's true" capacity, if it was there, easy to do, widely accepted? (Yes, fine, most rapes are committed by someone who the victim knows. There are still lots that aren't. Yes, fine, most single-gender spaces at most times still aren't great locations for an attack. There are enough that are.)

And if this does start happening, what, do you think the perpetrators will want their new access taken away? Of course not! But there's an easy way to help with that, too: just keep up the pretense while talking to the police. With luck, it'll never get coded as "male rapist invades woman-only space to commit rape". That might cause some sort of rule change.

Note that nowhere in here has an actual trans person ever appeared. In this hypothetical, the only people we are talking about are cis men, some of whom are rapists, cis women, many of whom are targets, and some rule changes. There are trans people too, of course, but nothing they do or don't do affects this picture. It's a completely cis/het scenario.

Something like this is, I think, what JKR may envision. It might be mistaken vision, but it is legitimately alarming. Places that once were private and safe now require constant vigilance and worry. Maybe that's the concern--I can't be entirely sure, but it seems highly plausible from what I've seen.

Most trans advocates, if they even consider any concern like this at all (usually I've just seen the person dismissed as transphobic, maybe with some disparaging comments indicating a belief that their concerns are expressed disingenuously), say things like, "No. No, this has not happened." But the rule change just happened. Isn't that the time to either anticipate problems or reassure people that the imagined problems are just that--imaginary? (On the basis of reasoning about life after the rule change, not direct evidence...because the evidence can't have been collected yet!)

This doesn't explain every aspect of, say, JKR's response herself. But I think it could explain rather a lot.

That said, I don't want to let JKR herself off too easily. She lends support to virulent anti-trans views waaaay too readily. But I don't think you've quite explored the possible mindsets behind her concerns as much as would be needed to engage with people with that type of mindset.

(Aside: if this type of concern is legitimate, given the shockingly high levels of sexual violence against trans women, I would think that the most worried group ought to be trans women. But that's another discussion.)