How do you figure that she is "rather obsessed with people who are either trans or don't meet her criteria of what a woman should be"? From what I have seen, I think it could possibly be reasonable to say that she is "rather obsessed with" the rights and needs of women, and that this sometimes leads to her making comments that may seem non-pandering unsympathetic from non-women-centered perspectives. I say tough; how many times do we normalize men and handmaidens being obsessively male-centered?
No idea re the nameless "football manager", but Imane Khelif is absolutely a legitimate (and huge) subject of discussion right now. If Rowling has been pointing out the negative impact to women of allowing men to participate in the specific sporting events that are reserved for women (BECAUSE women needed and fought for and gained the right to women-only events), then hell yeah it makes perfect sense that she tweets that. And the fact that tabloids retweet it isn't a negative (from a feminist perspective, mind - because this is a feminist board).
I would usually say "not my prob, go ahead" other than she could be putting trans women and other women at risk of harm with her tweets. Which are in response to situations which actually are, not "could be", putting women at risk of harm, such as males participating in specific sporting events which were previously reserved for women BECAUSE of the physical advantages given by male puberty, and because of the risk of harm to women participants.
btw where is her chum Bindel? Or is there a game going on where people go quiet from twitter so everyone guesses where they are? She's busy on the Ouija contacting feminists from the 1930s and 1940s, asking for advice about Sippenhaft.