I think the pro-birthers are for 2 things - more children in general, especially the real extreme ones who view Christanity as in some sort of existential war for supremacy by numbers against… any number of other groups of people (Islam, atheists, etc). Some literally view it as breeding an army, for which some view as a war for the world.
Second, the real extreme ones want to put the onus on women (as you point out, men wanting power over women), but they want us to do so through only having sex within marriage (preferably a proper Christian one), where sex is a sacrament whose only purpose is the production of children. Sex Ed and contraceptions are viewed as leading women astray and making them believe that they have an autonomy they don’t believe we actually have.
So yeah, this view really shouldn’t surprise us at all. And not all anti-abortion types believe this either. I’m sure there are some who are interested in sex ed and birth control, but they aren’t the vocal majority of the movement, unfortunately for them. These whack-o-dos are.
