without placing a value judgement on the outcomes, this assertion is only true if the population decreases overall, which doesn't happen in the countries with permissive immigration policies. other countries with low birth rates and strict immigration policies have different problems, like declining property values, difficulty paying for pension/retirement obligations, shrinking gdp (even if gdp per capita is increasing), etc.
Only if you're willing to starve the elderly.
Consequently, future generations will always be paying for present day late life social support.
Because of this, negative population growth requires higher and higher present day tax burdens to continue paying equivalent benefits.
And population economics are not something you can paper over, because they're by definition on the order of your entire country's economic output.
Now that some countries actually have shrinking populations, everybody switched from panicking about overpopulation to panicking about having not enough working population to support too many elderly people. Immigration is the popular way out, but somehow I'd expect this shouldn't have been such a big problem in the first place. A couple of decades ago, women barely worked (paid, that is; they were probably overrepresented in unpaid elderly care).
I don't think a shrinking population should be this big a problem for a country. Sure, it probably requires some economic choices, just like everything does. But especially for densely populated countries dealing with housing shortages, a lower birth rate shouldn't have to be a bad thing.
I always thought a low birth rate was a net-good: Something all countries are striving for. Like when your country drops below 2 births per woman, it has "made it" into the ranks of Developed Countries. Lower birth rate means fewer bodies to feed and a smaller environmental footprint. Sure, I guess it also means you have less "cannon fodder" both economically and militarily, but net net low birth rate countries are better off than high birth rate countries.