The time to obsess
Oh come ON.
JKR doesn’t “obsess about” Sturgeon, she resists powerful people who undermine women’s rights. None of this is about Sturgeon the person, it’s about women’s rights and how we can continue to have them when so many powerful people are hell-bent on taking rights away from women in order to give them to men.
As for “gratuitous cruelty” – have a word with people like “Sophie Molly” and “India” Willoughby and the rest of the women-hating crew. Seriously: have a word with them.

I bet Sturgeon’s never read what Rowling said about trans. That hardly qualifies as ‘gratuitous cruelty’ – or any other kind of cruelty.
Maybe Sturgeon needs to believe JKR obsesses about her. Maybe it gives her a high, knowing someone so famous noticed her.