> What is also funny is that settled immigrants often have more right-wing views about immigration than people born into that society.
> This doesn't appear to happen in the US
Yes, it does. It may happen significantly less in communities where the immigrants, their descendants, and others of their ethnicity are targeted, without selectivity by immigration status, for bigotry, including the highly visible Latino/Latina community (though, even there it still is a recognized and not uncommon phenomenon.)
It definitely happens among White European immigrants.
> It seems very odd to say limiting immigration is white supremacy.
Limiting immigration isn't, inherently, white supremacy.
But the people pushing it hardest in the US are, quite overtly, also pushing White supremacy and courting and excusing White supremacists.
> The US has a huge illegal immigration problem
The United States has had net negative illegal immigration for over decade, and the gross level has been in decline, too. It does not have a “huge illegal immigration problem”. And the illegal immigration problem it does have is in significant part because of stupid features of it's immigration system, like per country caps on top of per category caps, which creates decades long backlogs for legal immigration from countries with large numbers of prospective immigrants meeting individual qualifications, and stupid features of it's domestic policy that intersect with immigration policy, like the war on drugs and the resultant exportation of hardened, long-term US resident gang members to Central America that have contributed to the violence and refugee crises there.