Thanks for your thoughts. My gender is similar to yours, and I basically agree. I want to actually have abortion rights (and repro rights and bodily autonomy in general). Whether every single public communication is phrased in a gender-neutral or gender-inclusive way is of less importance to me. It's not a big personal deal to me if a "women's clinic" calls itself such, as long as they let me (a bearded person with male-coded ID and masculine name and pronouns) make an appointment if I ever want or need one and I don't have to give them "what is a trans man?" 101. Especially since there probably isn't a "trans clinic" that provides all the same services (e.g. abortion), I need access to the "women's clinic." Moreover, cis women need access to the women's clinic. If the women's clinic can't operate—at all, for anyone—because abortion is illegal, that is the giant problem about which we must all be concerned.
Somewhat analogously, I care little if social service agencies and hospitals have words like "Christian" or "Jewish" in their name, as long as they don't discriminate against anyone who needs their help. They have a branding challenge, and if they choose to live out that branding challenge while simultaneously doing the work to actually be inclusive, fine. They're picking their own language fight with the universe. If I'm not part of their organization, I'm not well positioned to have the argument with them about how they use language, nor does it really affect me as long as they would treat me equally. Do they know that non-religious trans people exist, and are they prepared to welcome those clients, because they've done that work? Yes? OK. Personally I'm not losing sleep over what is a non-situation for me.
When a trans person is denied abortion access because of ignorance, exclusion, etc., that's a different situation.
FYI, the Trans Journalists Association has an interesting recent Twitter thread: https://twitter.com/TransJA/status/1522215688080814081
They suggest that we can often rely on the terms "people seeking abortions," "abortion patients," "pregnant teens," etc. since those terms represent exactly that. Pregnant people and abortion-seekers have uteruses—we know that, so it doesn't need to be pointed out—and usually there's no need to identify or speculate about people's possible genders. A phrase like "women, nonbinary people, trans men, etc." might not add anything to the sentence. The association suggests it's accurate to refer to "women" when, for example, citing a study that only considered cis women, since we might not know if the study's findings would have been the same for trans men and nonbinary people. We can point out that a study of cis women produces findings only about cis women. In that case, the word "women" doesn't exclude, but rather highlights who has been excluded and therefore who isn't covered yet by our knowledge.
Similarly, I think, talking about abortion as a "women's issue" may sometimes be a way of acknowledging past discourse/realities and need not be prescriptive of future discourse/realities.
