Free market economics built America into an economic powerhouse, not nationalist bullshit.
Culture changes. Values continue to matter, particularly in the case of rational egoism and free markets. The people who stay and vote are the ones who are most attracted to here and who will adopt those values.
By generation 2, most offspring of immigrants (legal and illegal) are bilingual. Starting with gen 3, significant percentages don't even speak spanish. If you want a proxy for assimilation, there's a half decent one.
The fact is, I think citizenship should be hard to get and residency easy to get. Our legal system provides for that already; let's just use it
Go visit boston and you'll see a ton of IRISH PRIDE and Irish flags. No one goes whining that they're not assimilating. If they like Mexico they can go back to it any time, as you've mentioned; so it seems unlikely that people who just don't like the US would stay long term anyway.