Both sides do it, but that's a prime example of one of the popular ones from one side.
But if the dominant immigration flows to the U.S. were like Cubans in Florida (https://www.aljazeera.com/news/longform/2026/8/14/the-party-...), I strongly suspect we would be talking about how wealthy Republicans favor immigrants for cheap labor.
Although I agree with you that non-white immigration has had the effect of making the electorate more paternalistic, my guess is that most on the Left would be disappointed to learn that.
The pro-immigrant Left must believe that the immigration helps in the fight against oppression somehow or that it helps the immigrants while not harming the US. I think particularly that they reject (as racist) any suggestion that non-white immigration might harm or diminish the US in any substantial or lasting way.
Some of the comment on HN over the years claim that immigration is essential to keep the US from collapsing (although I don't know whether that view has any influence among Washington decisionmakers). If someone believes that US whites got most of their wealth and privileges by extracting labor from non-whites, then it starts to make sense to believe that the US economy is structurally dependent on exploitation and consequently needs a steady stream of non-white immigrants to extract labor from. (Maybe for example he believes that the children of immigrants are less exploitable than their parents because for example going to college in the US insulates a person from most forms of oppression.)