@baffledbunny
You could read what JKR wrote and form your own opinion but the basis of disagreement is, I believe, that she makes a distinction between sex and gender and makes a case that sex is important, in some situations in particular e.g. domestic violence refuges.
The misogynist / incel types you rightly identify as having infiltrated the pro-trans movement will not have this. They want nothing left for women, alone, at all. They promote the view that gender is always more important than sex, so if TWAW are women in terms of gender, they must always be treated as women in terms of sex too, so welcomed into all single sex services.
On the surface is looks like a silly muddle between gender (social and psychological) and sex (physical). Most people recognise that difference and would be astonished to hear that anyone honestly believes that people can change sex.
You'd think the TRA movement might be more concerned about changing the hearts and minds of the many people who don't recognise that gender really exists, or in explaining the importance and implications of treating people as their chosen gender, whether you believe in gender or not (and respectfully negotiating the boundaries of that practice, where it conflicts with others' interests), than in promoting the notion of this kind of magical transubstantiation.
But no, they'd rather go after a famous woman who is saying what most people believe - and in doing so, create confusion about what she actually said, plus a sense of jeopardy about anyone ordinarily saying ordinary things out loud.
The rest of your post is a good example of that confusion.
I know she's pissed off some transpeople by referring to women as having periods and not including transwomen (i didn't even know they did before then) and then I think she dug her heels in and refused to acknowledge that trans people can have periods too, hence the anger towards her. Is that correct or was there more to it?
Transwomen do not have periods. They have no wombs, no ovaries, no eggs. How could they possibly have a period? How could you possibly imagine that they could?
The distinction being made there (and I don't know if JKR actually said what you suggest she did) is either about whether only women have periods (some transmen would argue otherwise), or, as I suspect is the point, women being those people typically capable of having periods (thus trans-women are not women as they are not female. Woman meaning adult, human female).
That is what would upset people who want everyone to believe that TWAW, whether only in terms of gender, or additionally in terms of sex.