I just caught this--I don't know how I overlooked it before--and it is hard to overstate how misogynistic this attitude is. I really hope that this is just clumsy framing on your part.

Of course it's not inclusive when you use terms that some group hates!

Not even if the group that hates it happens to be (cis) women.

In case it isn't abundantly clear why this is a problem, the attitude that there's something wrong with pushing back against using terms that (a lot of) women hate is straight out of the right-wing toxic masculinity playbook. Even the justification for "birthing person" sounds identical to the right-wing argument for not respecting pronouns: "we're just tellin' it like it is, honey, just the medical facts here--you can give birth!". Are you going to add in, "and facts don't care about your feelings"?

It's a bad argument if the right uses it, and it's not any better if trans people or trans advocates use it. It's a bad argument because the problem isn't about a lack of accuracy, it's about how to formulate a cultural convention that displays respect. You can't dictate to others what feels respectful to them; you have to listen.

If you think that Ana is factually wrong that (a substantial number of) women hate it, that is a totally different thing and needs to be expressed differently. If you think that the term "birthing person" can be viewed as endearing and warm instead of an impersonal reduction to bodily function, make the case for that. But don't tell people that they don't have the reaction that they do, and don't tell people that it doesn't matter how it feels to them, and certainly don't think you can get away with it because they're just (cis) women.