I'm pretty certain the original article (which Rowling essentially laughed at) was the one being inclusive and using specific language.
The article was talking about how access to hygienic facilities and sanitary products continues despite the pandemic. This effects people who menstruate. The only inclusive and specific language to use here is "people who menstruate."
It would be incorrect to say "women" or "females" in this context. Not all women menstruate. Every if you want to be a trans-person denialist, not all people assigned female at birth menstruate. i.e. those who have not reached puberty, those in menopause, or those with amenorrhea.
JK was mocking the use of specific and intentional language. Her suggestion of saying "women" instead of "people who menstruate" was, at best, medically or scientifically exclusionary and inaccurate. At worst, she was intentionally casting aside transpersons from being part of the discussion.
ahh...
I might let it slide if it was in casual conversation - but the context suggests otherwise