I mostly agree with the idea that women in modern society are given general priority/encouraged to spend time with their children vs their male partners. What I disagree with is that this fact of modern society is biologically inherent to the ability to give birth itself. (Maybe related to biology in other ways, I could see an argument for hormone-based brain differences, but that wasn't what I was nitpicking in your first post.)

The multiplicity of options that women currently have (stay at home mom without shame and/or a career), is a result of how our society reacts to the biological fact of childbirth (among other things), not an innate result inherent to the fact itself. You suggest this yourself by saying "unless women are being enslaved or raped (not the norm in the first world), they control who gets to procreate." In other societies, the ability to give birth could result in widely different outcomes for women.

Additional edit: Even accepting your premise that women have more "bargaining power" than men in interpersonal relationships, that would again be a function of society's reaction to biology rather than biology itself. It's true that bearing a child to term takes several orders of magnitude more time and resources than the act of impregnating someone. But the idea that "limited resources have a price" and so the people who can become pregnant have leverage against the ones who don't isn't biological. Limited resources could just as well be seized and then redistributed by the government to the general population for free. The fact of childbirth is biological, but the reaction is societal.