Yea I'm not really fond of gender-neutral terms in this context. "Birth giver", "menstruator", "womb haver", "people/bodies with x or y" or my favourite "birthing bodies", it just feels like objectification and dehumanisation.

And once I saw a good point about people (like refugees) who live in a country where they're not native to the language. It's easier, for example, to understand that "if you're a woman, you need pap smears" than "if you have female reproductive system..."

It reduces women to their biological functions. I have nothing against gender neutral terms anywhere else, but these are pretty icky. Also you ever noticed no one talks about men as "penis havers"?