I think you're drastically underestimating how many people emigrate to America. As of 2018 44 million Americans were born in another country and immigrated here[0], representing one fifth of the world's immigrants. As a percentage of the US' population (13.7), this is very close to the all time high of 14.8% in 1890. Our immigration wave right now is damn close to the peaks of the Ellis Island period of Italian and Irish immigration in the late 1900s. And that's as a percent of the current population, by total numbers this is the biggest wave ever.

Japan is trying to increase its immigration for the same reasons I mentioned, and they're struggling for the reasons you mentioned. Only 2% of their population is foreign born[1], which is up from past figures, but still a drop in the bucket compared to the US system.

You're totally right that there's going to be political and cultural issues over immigration, there always is. But America has the political and social institutions necessary to maintain a high level of immigration over long periods of time, which can help damped the blow of fertility change even if it's insufficient to fully reverse the trend.

0 - https://www.pewresearch.org/fact-tank/2020/08/20/key-finding...

1 - https://www.oecd-ilibrary.org/sites/e025d47d-en/index.html?i...