> Is it all that repressed guilt from invading Indian lands or something?
I don't think you have to look that far back to find fairly convincing arguments that the US is the architect of much of its own immigration "problems." Most illegal immigrants come to the US from Latin American countries that the US spent a lot of time interfering with in very recent history.
Consider Guatemala. Democratically elected president overthrown with CIA support in 1954 so that US fruit companies could keep up their profits by exploiting people. The 1950s weren't that long ago.
Consider El Salvador. During their recent (ended: 1992) civil war the US funded the right-wing government that according to the UN committed 85% of atrocities during the war. The US government then refused to grant asylum and legal protections to refugees, which contributed heavily to MS13 forming in LA. The US then deported many of these gang members back to El Salvador, which did a huge amount of damage to their development after the war.
Wrt Mexico's gangs, most of the demand for their products come from US customers.
Wrt Venezuela, whether the sanctions are right or wrong there's no doubt that they're hurting people there economically.
Given what the US has done to these countries (and others) in living memory, I don't think we have much of a moral right to turn these people away. I've also never had a single negative experience (and many positive ones) with immigrants from these countries (I live in an area with many of them), nor am I convinced that they're even an economic drag on the country.