'.... we are a nation of immigrants'

Not really. We are becoming that, as the native population is swamped by the endless hordes colonizing us. More correctly, we are a nation of settlers, who created an entire new polity here on this sparsely populated continent. Immigrants came later, and in theory should assimilate into that polity.

There are multiple ways to debunk rationalizations that immigration in the 21st century is a good thing, but little to support it. 

You can use a cultural argument: prior to Hart-Cellar in 1965, immigrants were all European. Different languages, source countries, but all Western. Assuming all the other tribes assimilate equally in fanciful. And in any case, should we risk the destruction and dissolution of the US for the imagined benefits of this policy?

The most compelling argument of course is economic. Half of jobs are going away via automation in a few decades. Immigrants create a fiscal lifetime burden of $230k or more, and we are $20T in debt.