In answer to your question, EyesOpening.
"People who menstruate", "menstruators", "cervix havers", "chest feeding" etc. all reduce women to their bodily functions which is a bit offensive IMO.
It's also a bloody disaster when used in a public health context - which is clearer if you have English as a second language, or struggle with reading, or simply aren't familiar with the Latin-derived names for bits of your anatomy (including internal bits which you've never actually seen) - "If you're a woman, remember to book an appointment with your GP's surgery to check for cervical cancer" or "Are you a cervix-haver? Then go ..."
But worst of all it decouples our biology from our political struggles.
Abortion rights, for e.g. How come men are allowed to have "just wars" but "abortion is murder"? Because abortion is something women do. If it becomes something "some people do" abortion as a women's rights issue becomes decoupled from all the other issues in women's lives, like the fact that we are much more likely to be on the receiving end of male sexual violence, that we're economically disadvantaged because of our sex, that there's a power imbalance with relatively few women in positions of power - whether that's on the small scale within the average company, or on the large scale in terms of national governments. Our biology, our reproductive rights, our position in society - they're all tangled up together. Try to hide from the fact that it's women that need abortions and you're kicking the reasons why under the carpet, in countries like Nicaragua, they're denied abortions.
(And don't even get me started on the fucking Guardian re-naming "female genital mutation" "forced genital mutilation".)
In slogan form: if you can't see sex, you can't see sexism.
Oh, and that group of people who menstruate. Women. All of them. Some of them may sincerely wish they'd been born male, or "feel male inside", but every single last one of them has female biology.