I think you have misjudged the impact of LEGAL immigration on low skill wages in the USA.
Since 1990, America has issued 27 million Green Cards, mostly to low skill, low education workers who speak English as a second language.
That fact alone - completely independent of illegal immigration, completely independent of outsourced factory jobs - has crushed the wage scale for low skill American workers.
Yes, we need to reduce levels of legal immigration, too. But many of those “legal” immigrants came in as illegal immigrants, or chain migration to legalized illegal immigrants. But I’m only explaining my choice of words; I agree with you entirely.