If you want to provide appropriate care to people with wombs you don't want to filter based on what their drivers' license or birth certificate say, because those can not only defy your expectations, but they can also be wrong due to clerical errors or other factors.

I can't even humor your assumption that there are only two sexes and one birthing sex, because intersexed people exist, as do people with unusual chromosomes. Some of them have wombs and can give birth. Language like "birthing persons", while awkward, admits the existence of those people into a larger group without requiring even more awkward phrasing akin to "women, and men with wombs, and intersexed people with wombs".