Let them. There should be citizenship for anyone living or working in this country for six months regardless of how they get here. To do otherwise is to exploit people and subject them to man's inhumanity to man as we are seeing with recent administration policies. That and the precarity and exploitation foisted upon undocumented workers by US corporations using the threat of state violence while crowds call for blood.
A steady trickle of immigrants is much better because then it forces cultural integration, especially when the immigrants do not speak English. A flood of immigrants will lead to a nation divided among cultural lines without the possibility of integration because the cultural bubbles will have already been established.
We should focus on helping impoverished nations develop, so people don't have to leave from their birth countries to find comfort.
In the US there's really not any community of non-English speaking citizens, so whatever to your complaints about integration, the kids desperately want to integrate (and the parents usually agree).
And of course the whole thing where foreigners are scary is as much a modern panic as anything. In periods of pretty open immigration, lots of states didn't require citizenship to vote:
There are hundreds of thousands if not millions of Spanish speaking people and families living in LA, NYC, Miami etc. Many business in the area offer their services in Spanish, and there are enough entertainment options in Spanish that there isn't much of a need to learn English to get by day to day.
So first of all, the numbers on that page support my point without whinging over definitions.
But if you look at what I said, I didn't say anything about language spoken at home or primary language or anything like that, I explicitly said community of non-English speaking citizens, which pretty clearly means people that don't speak English anywhere.