When she initially got up in arms about the term "people who menstruate" she ignored one vital point. When a female starts to menstruate, along with when she develops breasts, she is developing the secondary characteristics of her biological sex, but a female doesn't become a woman until she reaches the age of majority, which is 18 in many countries and 20 in Japan.
If we want to see evidence of they win and you lose, look at the nonsensical arguments put forward by opponents to marriage equality. A group in society gaining equal rights does not mean a loss of space, it means a loss of privilege or having to share your privilege with a group that didn't have it. If you want to see a genuine threat to democracy, look no further than the orange haired individual who protested that "the election was stolen!" He claimed that if elected in 2024, he will remove all of Joe Biden's humane (he said they were cruel) actions to protect trans folk and would prevent access to cross sex hormones until a person is 26!
One of the things that needs to be remembered with transwomen in women's sports is this. Nothing is yours until it's yours. If you remember Charlie and the Chocolate Factory, the first four children to win golden tickets were a greedy boy who ate lots of chocolate, a spoilt girl whose wealthy parents could buy her anything she wanted, a girl who chewed lots of bubblegum and a boy who watched TV and was interactive with the TV. When Charlie Bucket found 50 pence in the snow at the time that his family was beginning to starve, he bought a five-penny chocolate bar and wolfed it down, and decided to have one more and found the last golden ticket, a boy was angry and said that he'd been buying 20 bars of chocolate a day and yet he missed out to a boy who was an outside chance. These women who say, "Oh, so and so should be on that podium," ignore something. If an athlete does really well and an official says, "Piddle please," and they find performance enhancing drugs or masking agents in their system, they are stripped of their medals, and rightly so, but a transgender athlete has to meet certain levels before they can compete as a woman. So, if a transwomen meets the criteria and qualifies, and has not cheated, then complaints are sour grapes!
Where JK Rowling is beyond the pale is in suggesting that transwomen are not entitled to the presumption of innocence when it comes to sexual crimes. Hell, to take an extreme example, the police had all the evidence that Fred West was guilty of the House of Horrors Murders, even he was entitled to the presumption of innocence until guilt was proven!