She's asserting the rights of women to have female-only spaces, and to not be linguistically erased. That's not exactly "spewing hate" is it.
Female only spaces would include all females. Not whomever Rowling decides has sufficient quantities of the essence of what she defines as woman.
I wonder if you did not know about that, or if you think that that person should share dressing room and showers with biological women.
If you didn't know about that, maybe now it's simpler for you to at least a bit see how JKR is thinking?
https://www.theguardian.com/uk-news/2023/feb/28/trans-woman-...
(Just to avoid misunderstandings: Such events are of course very rare, the trans people I know are kind and lovely people)
https://fairplayforwomen.com/transgender-male-criminality-se...
More research is needed, but I think it's not an unexpected conclusion. I mean, the only difference between these men and other men is that they're calling themselves women. Most of them keep their genitals intact, and most are driven to identify as women due to a paraphilia.