The phenomenon Spandrell describes is a concern largely independent of whether the total population is increasing or decreasing.
Every advanced country has built a system for vacuuming up the most talented individuals from the countryside, and giving them important jobs in rat-race cities, where (for various reasons) they will have fewer kids. To counter this, you'd have to draw from somewhere else equally talented newcomers, and maybe a few countries can.
Humans in (actual or perceived) stable social and economic environments with strong extrafamilial support systems adopt low-number, high-investment K-strategy, humans in (actual or perceived) unstable social and economic environments with weak extrafamilial support systems adopt high-number, low-investment r-strategy.
> kids are very costly in today's society,
Kids aren't (generally) nearly as costly for the people having lots of them in our society as they are for those having fewer, as a direct result of different child-bearing approaches.
> The answer to me is simple: we need to outsource child-rearing to the State, as shown in the novel "Brave New World" where children were made in factories and then raised by trained professionals.
In the short term, immigration works. If the whole world is ever developed to the point where K-strategy dominates to the point where maintaining population is a real concern, then, after the giant global celebration we should have for that accomplishment, setting economic incentives so that people with means to raise multiple children at the level society as a whole considered appropriate investment levels also have a rational incentive to do so isn't particularly hard. It's a problem we haven't solved today because we aren't anywhere close to having it, not because it's even a little difficult or requires some kind of drastic reform like you describe.