"There's nothing wrong with using terms like cervix haver! It's just more inclusive!"
No, it's not. It's also so incredibly confusing for the most vulnerable women who are already at risk for not getting appropriate medical care.
I work with homeless people. Most of my clients are drug addicts. A lot of them have some sort of developmental disability, others have had head injuries that impact their cognitive function. Some of them have psychotic disorders that make it hard for them to recognize what is and isn't real. Most of them have intense trauma that impacts the way they interact with the world.
Most of my clients are not functionally literate. I have an easier time counting which clients have completed high school than those who haven't. These women have very limited access to the internet and phones, and when they can access the internet, their options are still limited because they cannot properly read.
So if you change "women's healthcare" to some bullshit like "cervix havers healthcare" or "menstruaters healthcare" and put that on the posters instead, these clients don't give it a second glance. They don't know enough about their own bodies to recognize a term like "cervix" including what it is or if they have one. They cannot understand the nuance that those terms apply to them.
I work with a woman who has severe brain damage from her lifetime. Abusive childhood, early drug use, and then life on the streets has left her memory and mind severely impaired. She needs her case manager to remember her bank pin for her, because a four digit code is too much for her. She struggles to read words like "would" or "shelter" and has a limited understanding of her own body. She has liver issues that she keeps identifying as kidney issues because she doesn't really know or understand the difference between the organs. Do you think that she would hear an advertisement for "cervix havers" and recognize that it applied to her? No, of course not.
I work with another woman who has a severe developmental disability and a psychotic disorder. Abusive childhood with suspected child sex trafficking. She's had two children already, neither of which she has custody of. She can recognize what pregnancy is but not the finer details. To her, the baby grew in her belly. She could not make a uterus if you showed her a picture diagram of the female reproductive system. If you put up a sign for "uterus owners" and their healthcare, she'll walk right past it. She won't recognize that it applies to her.
Talk to anybody who works with the homeless, getting these folks connected to healthcare is already difficult without throwing in nonsense like so called "inclusive language" to make it worse. Doctors already have a hard time conveying medical instructions to these patients without having to trip over themselves by following the TRA approved script and muck things up more.