You definitely need to be female to give birth, regardless of whatever gender identity label you give yourself (e.g. "man" or "non binary").
https://pmc.ncbi.nlm.nih.gov/articles/PMC8762399/
tl;dr: All boy moms will have some proportion of XY cells lodged in their bodies until they die. And this can result in them showing up as genetically male on a DNA test, simply because they once gave birth to a boy. When you get down to a cellular or molecular level, notions of "sex" or even "individuality" are far more blurry than our mental models admit. Sorta like recent findings that by cell count, humans are only about 43% human, and the rest is the bacteria making up our microbiome, and that our microbiome can have significant impact on our behavior.
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".
My point is, our interpretation of these observable biological processes and facts, and our response to them, is socially constructed. There are many Doctors and Scientists who do very much agree with this complex view, just as I'm sure there are those who believe in a strong binary like you. I see the complex view here as much more useful, especially in navigating things like human dynamics and healthcare needs in ways that benefit us all