I'm pretty sure the horrific transphobia in Rowling's essay is things like this:
"hitherto totally unfeminist older lady who’s vowed never to visit Marks & Spencer again because they’re allowing any man who says they identify as a woman into the women’s changing rooms"
"reasons for being worried about the new trans activism,"
"I’m concerned about the huge explosion in young women wishing to transition and also about the increasing numbers who seem to be detransitioning (returning to their original sex), because they regret taking steps that have, in some cases, altered their bodies irrevocably, and taken away their fertility. Some say they decided to transition after realising they were same-sex attracted,"
"I’ve wondered whether, if I’d been born 30 years later, I too might have tried to transition. The allure of escaping womanhood would have been huge."
"A man who intends to have no surgery and take no hormones may now secure himself a Gender Recognition Certificate and be a woman in the sight of the law."
"I’ve read all the arguments about femaleness not residing in the sexed body, and the assertions that biological women don’t have common experiences, and I find them, too, deeply misogynistic and regressive."
" ‘Woman’ is not an idea in a man’s head. "
"When you throw open the doors of bathrooms and changing rooms to any man who believes or feels he’s a woman – and, as I’ve said, gender confirmation certificates may now be granted without any need for surgery or hormones – then you open the door to any and all men who wish to come inside."
These types of things are read as horrifically transphobic by trans activists because they involve the idea that you can't change sex, that saying you're a woman doesn't make you a woman, that single-sex protections and services should be upheld, that making everything unisex is often undesirable and dangerous for women, that gender identity is largely based on sexist ideas, and that for many these problems with their "gender" are temporary.
The reason they don't want to point to these quotes from the essay to back up their points is that they're aware that these are normal statements with which most members of society will agree with and not find objectionable.