Good to see you back @baffledbunny I feared we might have put you off by answering your points directly. Though you don't sound like someone easily scared away from discussion.
On your 'taking sides' point, I'm afraid I do find it very primary school and immature for people to jump to one 'side' or another without assessing the evidence themselves. I realise that people are tribal and will naturally flow with the cultural norms of their social group. I see that as a more passive phenomenon than actively choosing to 'take a side' though. What happened to 'I don't really know much about that.'?
Of course many people on here plainly have taken a side. But they've done that as a result of assessing the evidence (often having started out with quite different assumptions), not in advance of assessing that evidence.
On the periods / refuges point, my point is that they are two sides of the same coin. 'Being kind' and including transwomen in the definition of woman, results in non-trans men gaining access to women's refuges, prisons etc.
Transmen are female and they know this about themselves, as well as the extent to which they have or haven't amended their own biology, so it is not always necessary or accurate to say 'women and transmen' when discussing female biological processes. When talking about biology and medical issues, woman means adult, human female, not gender identity.
Confusing and conflating the two in a medical context causes more problems than it solves - like the high number of women who don't know enough biology or English language to know that they have a cervix, so don't realise that they need to book a smear. That number, of women who could die of cancer as a result of the confusion, is much higher than the number of transmen who might feel uncomfortable at being reminded that they have female biology.
But as ever, the demand for linguistic change isn't about transmen, it is about transwomen. The desire to describe adult, human females as 'menstruators' etc comes from transwomen who wish to take ownership of the word woman, to describe women as a mere sub-set of womanhood and to be acknowledged as women in all circumstances. The TRA backlash against JKR is based in that desire. So, the 'menstruator' wording that JKR was objecting to is directly about men gaining access to women's safe spaces.
Interestingly, when you say I'm relieved somewhat to realise that she is not anti-trans (I abhor outright bigots) and I agree with her on the need for women to have safe spaces of their own, including in prisons. you cast yourself as a TERF, in the terms used by TRAs.
Congratulations and welcome!