I think this is an oversimplification. History has shown that as soon as a country is developed enough that children start increasing the family expenses rather than decrease them (I.e. helping out with the farm, or whatever the sustaining family business is, but in developing countries this is overwhelmingly agriculture) the pressure to have children slacks off to a large degree and becomes more of a luxury. So it’s just a byproduct of industrialization.

The US is actually better off with replacement rate than a lot of countries that have industrialized since them because of the way it happened and the wars that were fought. More rapidly-industrializing countries (China, Japan, a few other Asian and SA countries) have way shorter runways despite industrializing much later than the US. And those with one child policies really just made things worse for themselves.

A very large part of what the future is going to look like in my opinion is how different countries are able to grapple with this issue and come up with solutions to the problem of a large aging population and a service, hospitality and medical industry with not enough bodies.