I have a similar theory, that desire to procreate is heritable, in a way that was previously inextricable from desire to have sex. With easy birth control, those desires can now be fulfilled separately. We're still working through the mass die-off of the genes that mostly just wanted the sex half of the equation.

In a few generations, most everyone alive will be the progeny of people who really wanted children. This is probably heritable and will probably stabilize birth rates.

Maybe. I think the difficulty is that in a place like Korea, the dependency ratio will become extremely high, and so taxes will have to go up sharply. Most voters will be retired and so will vote for the few young people to pay them. This will lead the young people to emigrate unless they’re prevented from doing so.