South Korea's birth rate is indeed dangerously low and threatens their economy. If they cared they could do something about their extreme workaholism and cost of living.
As far as immigration destroying the culture: it depends on the rate. A healthy but limited rate of immigration revitalizes cultures, but it's pretty clear that sudden mass immigration is dangerous and causes a lot of problems and backlash.
Speaking of fixing problems... if our billionaire class who seem so worked up about birth rates wanted to do something about it they could also address the most frequently cited problems preventing young families from forming. Like housing prices.
If we really get into a situation where dangerously low birth rates are threatening the economy and social stability, the boring solution is to do what we do for other goods and services that the free market isn't adequately providing: directly subsidize it. Pay people to have kids. The easy way would be a payment or tax rebate to married couples with 0 children to have one child. But that's the boring solution that doesn't involve massive reactionary cultural transformations, so the right-wing "problemists" won't like it.
Edit: by problemism I mean people who use a problem as a lead-in to push only one solution, and they're not actually interested in solving the problem unless it's their solution. An example from the left side of the spectrum would be Marxists for whom any solution to climate change that doesn't destroy capitalism is unacceptable, or anarcho-primitivists or anti-tech trads for whom any solution that doesn't involve shutting everything down is unacceptable. In reality the solution to climate change is to emit less carbon, and physics doesn't care how we do that or what political system we are using.
There's a lot of right-wing problemists for whom any solution to excessive population decline that doesn't involve taking rights from women and re-introducing reactionary theocratic or nationalist structures isn't acceptable.