I'm back after my initial rant. Just to offer a comparator from another language. Just discussed with a Finn to confirm.
In Finnish words for lady bits you can clearly see three eras of word formation.
In the oldest you find "emätin" (vagina). That's a clearly Finnish word, which somewhat (at least to me) connotes "instrument (-in) for causing (-t-) motherhood (emä)". I like that.
In the newest you find direct transliterations like "vulva" or "klitoris".
But in the middle period you find terms like "häpy" (vulva, or genitalia) and "häpykieli" (clitoris - vulva/genital-tongue).
But "häpy" is also "shame". This won't be a coincidence - clearly someone centuries back translated the Latin anatomical term "pudendum", based on its other meaning. This was quite common back then - scholars going off to foreign lands, learning Latin, Swedish or German stuff and translating it "literally".
So, the question is, how would the anti-sexism anti-shame scholars fare in Finland? It's a direct analogue - it's a word meaning shame for genitalia that is primarily used for female, but not exclusively.
But the difference is that it is obviously shame. Exactly the same word. Wiktionary's example sentence "Eikö sinulla ole yhtään häpyä?" - "Don't you have any shame?" could equally be "Don't you have any vulva/genitalia?". You would generally assume the first meaning though. 
In a sense, unlike English, the collision is so obvious, you barely notice it. (Like "can" in English - "be able", or "tin"?) That the two meanings of "häpy" might be related rather than coincidence might not have registered.
Asking a Finn, she just laughed at the prospect that someone might try to abolish "häpy"-related anatomy words. Not something that would have occurred to her.
But that said, having heard so much other nonsense, she could imagine someone might hear of this daft idea from English speakers and have a go. It might be possible given that there are probably other replacements - "häpy" itself is a bit old-fashioned - but it would be far more of a struggle, as it's not as niche as "pudendum". You would not just be attacking a couple of anatomy books.
She's certainly heard gender loonies complaining about Finnish's lack of sexed pronouns, so wouldn't put it past them... That causes trans people to be "misgendered", and doesn't give an inroad for pronoun circles and demanding neopronouns.