Birth rates are down worldwide except in Africa, and if I had to forward look based on current developed world policy (Japan, US, Europe migrant policy), I don’t think the US is going to drastically increase immigration quotas. Again, like Japan. Older citizens will want to keep their culture static, and they weight that higher than economic growth.
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.
It won't though. The UN's models on the area say that world population will continue to grow (population has momentum) until the end of that century, at which point population growth will be very close to 0, with an estimated population of 10.5 billion people. So even in total numbers, we won't run out of people anytime soon.
Even when global populations begin to fall, we can still keep immigration up. There's nothing at all inconsistent with falling populations and immigration. Economically there might be a lot of pressure for people in places with falling populations to emigrate, because of the economic effects of sudden population drops.
