Yet, the right enjoys that cheap labor whilst also keeping demand for jobs high (wages low). Odds are those "undesirables" won't achieve the American Dream™ and thus won't behold the GOP.
Great system yes?
You've hit on why the republican Congress wouldn't do anything substantial to stop illegal immigration. They like a supply of hard-working people, who don't mind doing hard and dirty jobs.
It's just a political ploy to hook people foolish enough to buy it. It's why republicans were generally in favor of the wall; it won't work.