The US was very good for a very long time at integrating immigrants. It should continue that tradition and work even harder at it.
I believe it was some Republican president who said something to the effect of “if you move to Germany you may be a citizen but you are not a German… But if you move to America and become a citizen you are an American”.
It’s worth noting that not all advanced societies have fared as badly as Korea and Japan. Scandinavia for instance is below replacement but not nearly as catastrophically as Korea. It’s possible that a bit more policy tweaking and more productivity=>leisure time could get them back to a replacement rate.
The US was historically rather hostile towards new waves of immigrants in practice, treating them very much like second class citizens (Irish, Italians, Latinos etc), effectively pressuring them to assimilate by becoming "more American than Americans" to avoid such attitudes. One can argue that the system kinda sorta worked in the long run, but I don't think it makes it worthy of emulation.