This has been discussed on the feminism board as well.
Some of the points I made there:
A VICE article about the Eve appeal survey
In a recent survey of 1,000 British women, 44 percent were unable to identify the vagina on a medical illustration of the female reproductive tract. Even fewer were able to identify the vulva, with 60 percent failing at this task. Overall, only one third of the women questioned could correctly place the six labels—vulva, vagina, cervix, uterus, fallopian tubes, and ovaries—on the diagram.
...
"The lack of basic knowledge about the female body is extremely worrying," Tracie Miles, a specialist gynecological cancer information Nurse at the Eve Appeal, said in a press release. "How can we expect women to know what to look out for in terms of unexpected changes in their vagina or vulva, or to be aware of the signs and symptoms of a gynecological cancer if they're not body-aware?"
As a woman, I find it dreadful to have my ability to refer to myself as my own sex erased. I want a collective noun that describes my female biology as a whole, as a person of my sex, especially in medical contexts that refer to reproductive organs where this actually matters! Woman has always worked adequately. I have no way of seeing my uterus or cervix, as they are internal organs, so how can I know if I am a "uterus-bearer" or "cervix-haver"? Equally, I think it's dehumanising. I wouldn't want men to be referred to as "vas deferens-havers" or "people with seminal vescicles."
Do I object to "chair-person?" No, because taking on the role of chair has nothing to do with biology, so there is little need to point out reproductive differences in this terminology.
Do I object to "ovary-bearer"? Yes. Because then you're referring specifically to the female sex, likely for reasons that have to do with female biology as a whole, but using language that reduces me to an organ and making things needlessly convoluted. How do I know I have ovaries? It's not like I can just look down and check...
The half of the population who need to come for screening tests was born female. I have never seen a collected group of women who have had hysterectomies protest against the use of the word woman! Because they know this is more important than being included or excluded by terminology!
If I have to compromise then "women and other people with cervixes" will be acceptable.
We have a perfectly good word, "woman," to denote adult human females.
This is just over-complicating everything!