Oh but this is exactly what I'm trying to explore with the South Korea specific example. By 2100 they are predicted by the UN* to have 1/2 of their current population. This remaining bunch will be mostly very old people, the fertile/working age fraction will be like 20% supporting 80% people too old to work or reproduce. This is already a never seen before situation (in peaceful times, at least), which you describe as "numbers tend to move back up, or at least stabilize", like we've seen this many times before, but we've never seen anything like that. Also, I'm having a very difficult time imagining a half-a-century long, continuing freefall in fertility suddenly reversing in another 5-8 decades and them all of a sudden starting to breed like rabbits in 2100, after they lose half their population. Even just to stop the freefall they would have to double the current fertility rate just to not go completely extinct around year 2150.
* https://www.bbc.com/worklife/article/20191010-south-koreas-p...
When your main home + a vacation home cost half of what your main home would have costed 3 decades ago, and you can afford them when you're 25, that's a huge relief right there.
The big question mark to that is that the entire economy doesn't collapse and also takes down salaries with it. But South Korea is quite export oriented so they're better positioned than some other countries.
If you look at South Korea's population, it's still being going up:
https://en.wikipedia.org/wiki/Demographics_of_South_Korea#/m...
(Edit: Actually in 2020 it went down for the first time in recorded history, it seems)
Give it a few more decades, when people actually start dying off en masse (because we haven't really found a way to reliably have masses of centenaries living), and then you'll should see the population dropping. Unless they increase immigration.
What I'm saying applies once the population actually drops by a large amount.