Even a birth rate as South Korea's does not mean the population will disappear over night. It will shrink. It will mean things will change. Infrastructure will be overprovisioned and housing will be cheap. It will mean other things will be prioritized by politics (such as kindergardens and work life balance).
In any case it won't be a catastrophy as life in North Korea.
A good analogy might be the Black Death: it didn’t destroy Europe, it changed priorities, freed the serfs, started valuing labor more, and ultimately led to a stronger Europe in the future.
If we consider that the world had 1bn people a bit over 100 years ago, it puts many things in perspective. We have generations of time to turn things around if we think it is necessary.
No, the Black Death is not a good analogy, because it killed all ages indiscriminately.
Our "plague" "kills" the young and productive - by their never being born in the first place. We are headed for something we have never seen, ever: a society dominated by old people in pure numbers terms.
The worry is that it is a design for stagnation and decay rather than greater strength. (There won't be money or people to maintain infrastructure because the elders will demand healthcare.)
I don't know what will happen and nor does anyone else really.
It killed all but made people more valuable and, more importantly, valued in general. If kids become more rare, we will probably start treating and nurturing them better. We complain about childcare costs now but if kids are few enough society might just happily take up the bill.
If only it were that easy. Austria and Switzerland have nerly identical birth rates, but in one, childcare is almost free and in the other, it is very expensive.
South Korea and Hungary (among many others) have tried paying people to have children, without success. It pulls a few births forward in time, but then the birth rate declines again.
In any case it won't be a catastrophy as life in North Korea.