> What's going to stop them from doing it when the birth rate becomes a problem? Almost nothing
Their birth rate is already a massive problem. The South Korean government already acknowledges this is a crisis, it's just that the measures that are politically/socially viable just don't cut it, and Koreans seem unwilling to consider more drastic measures. But the problem is already here, and acknowledged, and already impacting the population of South Korea (there's apparently a "loneliness epidemic" going on already).
Because of the shape the population pyramid takes (more old people than young people) once it reaches the tipping point, which in South Korea it already has, there's no going back. No matter how they try, they simply don't have enough young people to revert it anymore.
> If you organize immigration of labor, then not so many immigrants will be needed
This is not (just) about labor, it's about population decline. Even if Koreans dedicated themselves to having more children, it wouldn't be enough anymore. They are beyond the tipping point. They would need massive immigration to live there and have children there and effectively become "the new Koreans"... and this is obviously unpalatable to many.
I encourage you to read on this. Do not debate me: I'm not the expert here!