The argument is not implausible. Integration is important. When the citizens of a country view themselves through a tribal lens, it can be very hard to get important things done, and the country can become dysfunctional and - eventually - poor. In the past, America has done well at combining many disparate ethnicities - Irish, Germans, Italians, Jews, Greeks, Poles, etc. There's plenty of reason to believe that this is happening again, with the mostly Hispanic and Asian immigrants of the recent wave.
But there's an argument that we need to speed this process up, by pausing immigration. Without a pause, restrictionists say, the phenomenon of "replenished ethnicity" might keep Hispanic and Asian people feeling like "permanent foreigners" for decades, leading to tribalized politics and social strife. Only because we paused immigration in the past, they say, did we managed to integrate the previous waves.
That's not implausible, but I think a closer look at the history of U.S. immigration shows that past restrictions were not as important as many believe. Here, via Natalia Bronshtein, is a graph showing the history of U.S. immigration by source country. I've annotated the graph with some important events:
The things that stand out most are 1) the big pause in the early middle 20th century, and 2) the big waves in the early 1900s and late 1900s/early 2000s. The y-axis is in absolute numbers; in terms of percentages of the U.S. population, those two waves were about equally big.
One thing you'll notice is that there was no pause in the 19th century. Despite big waves of anti-immigrant and anti-Catholic sentiment, immigration was not banned and didn't halt. An exception was the Chinese Exclusion Act of 1882, but this didn't affect the biggest waves of immigrants coming in at the time.
But despite the fact that there was no pause and no ban, and despite the fact that Irish and German immigrants kept coming throughout the 1800s, immigrants from these countries integrated quite effectively into American society.
Another thing to notice is that when the big immigration restriction was enacted in 1924, immigration had already fallen substantially from its peak about 15 years earlier. The law was probably important, but maybe not as important as its fans think. I bet the Great Depression, which came just 5 years later, and WW2, would have been almost as effective in keeping immigration low.
Also note that immigration had started increasing substantially well before the 1965 law that loosened official controls. The 1950s were a time of rapidly increasing immigration, despite the legal ban. Nor was the 1965 law change immediately followed by a trend break; immigration increased steadily, but didn't really explode until the 1990s.
This doesn't mean that laws don't have an effect - the Simpson-Mazzoli act, commonly known as "Reagan's amnesty," was followed by a surge in Mexican immigration (and even more that was undocumented, and not on this graph). But overall, most of the ups and downs seem to correspond to economic booms, busts, and wars rather than to U.S. government policy.
So fans of the 1924 immigration restriction should rethink their understanding of history. Economic factors were probably just as important as laws in determining immigration levels.
Another important observation is that country-specific immigration booms all seem to end on their own. Irish and German immigration trickled off around the turn of the 20th century. Italian immigration experienced a short mini-boom after WW2, but never came close to regaining its previous levels. The Austro-Hungarian and Russian booms were short-lived, one-shot affairs.
Should we expect the Mexican boom to end similarly, on its own, without government controls? Yes. In fact, it already did end, at least a decade ago. It's done, finished, over, kaput:
More Mexicans are going back to Mexico than are coming in. Mexican immigration basically halted sometime in the 2000s and went into reverse. And yes, that includes illegal immigration, which has been negative since the Great Recession.
The Mexican Boom is done. The Hispanic Boom as a whole is not quite finished - Central Americans and Caribbeans continue to come in, though at a slower rate than before. But these are trickling off as well.
As of now, the main source of immigration to the U.S. is Asian. Asian immigrants are expected to surpass Hispanics as the largest foreign-born population in the country by mid century, unless Trump or other leaders block Asians from entering.
So the fears of "replenished ethnicity" keeping the American population from integrating are, in my opinion, overdone. Immigration booms end on their own. The new immigrants don't come from the same places that the old ones did. There is, therefore, little danger that allowing continued immigration will put us in danger of tribal balkanization.
For a more in-depth post on this topic, see this by Lyman Stone. Most of the conclusions and points are fairly similar, but there's much more theory and data.