Honestly I think 'women's rights' is the most accurate myself, not because trans men and afab nbs won't be effected by these laws but because by and large the people making these laws don't think that trans men are men or that nbs exist. To them those people are just women in denial. And while there's an argument to be made that using 'women's rights' just reinforces conservative transphobia, my counter argument would be... there are better ways to fight transphobia than taking a beef with women who are scared of what's going to happen to them if these laws stand or Roe gets rolled back.
When we qualify every breath with 'not just women but also ********' or use muddy language ('people with uteri,' 'reproductive rights,' etc) it creates a muddy message that isn't clear or compelling to fence-sitters, doesn't convey the root of the issue, and one that anti-choicers can use as a distraction. See: "The left are the REAL sexists, they think woman is a dirty word!" and "These people are loony and think men can have babies! You shouldn't trust anything they say!". Women are what this legislation hopes to control, anyone else who happens to own a uterus is unfortunately caught in the crossfire for this one.
I tend to still use this language in left-leaning spaces (especially online) because, despite others in this thread saying they've never encountered it -- I've seen a lot of grilling on facebook, twitter, tumblr, at my university, etc. over the years regarding language and specifically how language should be framed around female reproductive issues. So much so that it's just easier to twist my tongue or make all my paragraphs x2 as long as they need to be because I need to insert 10 qualifiers to every one of my statements than deal with the backlash for simplifying all that down to 1 term. Leftbook groups will even go wild and ban people for being transphobic if they use terms specific to women in this context.
I remember this being especially big during the whole pussy hat thing...which was never intended to be transphobic and was a play on Trump's gross comments (Trump didn't say 'grab the gender ambiguous individual by their pubic area' he was pretty damn specific) + what we all knew would eventually happen to our reproductive rights with him in office. From the backlash in leftist circles and LGBT think pieces posted around at the time, you'd think that trans folks were seriously offended by it, but the ones I know IRL were mostly fine with it and a few of my trans friends even donned them and went to the march *shrug*.
The attitude definitely exists online though, and the attitude is prevalent enough that a lot of pro-choice organizations have changed their official language around the issue. My general observation is that online I *rarely* see this come from trans men, but I don't see it come from cis men very often either (at least not genuinely). I hate to point fingers, but my observation is that online, in leftist circles/groups, it's mostly nbs, trans women, and well-meaning cis women who do this.
I recently saw a tweet passed around that I was certain had to be a sock owned by a cis alt-righter guy which said something along the lines of 'when discussing reproductive rights, cis women should remember that abortion is a privilege'... I did digging on the username and nope, it was really a trans woman, a popular one at that. I'm not a TERF but tone-def comments like that are how I imagine TERFs are born. It was infuriating and makes me want to throw up in my mouth a little just thinking about it. The worst part is that you can't even point out how wrong that statement is without extremists dogpiling you. Now, I don't think it's usually purposefully malicious/misogynist coming from them, I understand that they're attacked a lot when they don't deserve to be and I get that folks don't wanna feel erased, but... goddamnit it's frustrating.
Especially right now while all this is materializing. We're about to lose our rights. A lot of women (98% of the victims will be women) are going to die if we don't stop this from happening. Many more will suffer, many children will suffer. I don't have time or energy to deal with the leftist language police especially when the language police largely stand for the same thing (abortion and birth control should remain legal, safe, and accessible) and our common enemy has everything to gain and nothing to lose by this level of pedantry and infighting. Can't just this one struggle belong to women?