> I think it's more likely, drawing from biology, that we end up at a stable global population level without having to worry about moving backwards along the metrics of education, income or contraceptive access.

There's absolutely no inherent equilibrating force that will stabilize global fertility rates at replacement. Many countries have blown by replacement (the USA included) and continue on a downward trend year over year.

And if even cultural norms were reversed to pro-birth, it wouldn't be enough to reverse the trends, as the decline is compounding, and the increase of average age produces other complications (hard economic issues for starters, making people even more hesitant).