The delay between retirement and death mean people are spending their saving. So they will still retain the same purchasing power as long as their saving is sufficient, together with government account of relevant national governments or similar entities that pay for welfare that support such people.
Reduced productivity is not too much of a problen as the consumption power can be spent across national border.
Indeed, when you calculate the GDP per capita nunber itself, every single people leaving the work force would mean the number reduce, but does that actually matter to genarl population? It could be a concern if most elderlies are dependent on financial support from younger generation, but the US does not appears to be the case?
In the US it is actually even more the case as most retirement funds and savings are bound to stock markets, especially the domestic one. So if companies shrink as people retire and governmental policies do not allow to replace the workforce with immigrants, the output of the companies declines, reducing the return on investment on the stock and therefore reducing the stock value as well as dividend pay outs, reducing the savings/retirement funds of the elderly. This directly hits their spending, which in return reduces demand and GDP.
My understanding is many of those funds and savings target international market, even if they invest in domestic market, most of the significantly tracked stocks on the US market are those that have global workforce and global market.
Also, as operation efficiency worsen, some companies will ultimately be eliminated from the market, leaving behind stronger ones, as have observed during the pandemic.
Thats all true, the problem is that the invested part in companies overseas, while in theory staying stable, does not benefit the US at all except for the pensioners. Whats even worse, if funds start to divest in the US and flee the country overseas, US companies will go bankrupt. It is also not beneficial for the US if a company reduces jobs in the US and builds them somewhere else, because this productivity does not count towards US-GDP. So GDP will be reduced and jobs go somewhere else. On a globalist and neo-liberalism scale this is a good thing though, as it eliminates the need for high payed (expensive jobs) in the US and creates jobs with in total higher productivity in India for example.
Now this will reduce immigration because it makes the US a country not worth moving to because there are no jobs and no wealth (distribution). So while stock holders (and in this case pensioners) have a good live, normal people lose their jobs, cant build a pension fund and become poor.
EDIT: Thats why it is important to have inward and domestic investment, to boost and stabilize the economy. Countries with low inward and domestic investment go bust:
Just to highlight the effect of stagnating population and GDP growth:https://www.google.com/search?q=Japanese+GDPhttps://www.google.com/search?q=Japanese+population
As you can see they are facing a massive problem: Worker-Shortage, the economy is in huge trouble. The Yen is tanking, industrial output is sinking, while government debt is exploding. Its a ticking time bomb.
Now translate this onto the US, the only way the US can service its debt is by improving economic output to get more tax income. Its all linked together and the US is in no position to limit immigration at the moment except people are ok with reducing the country to dirty poors 99% of the population and filthy rich 1% of the population.