Immigration Debate

It is said that high end immigration (doctors, scientists etc.) is fine, but that the problem comes from "low end immigration" (the people who mow your lawn or wash your dishes.)
But we will get the fruits of high end immigration from wherever it comes. For example, a cure for certain diseases will fairly rapidly be available worldwide (slowed down somewhat by bureaucracies.)
But "Low end" immigrants cannot be replaced; someone has to be here to mow your lawn or wash your dishes. Which allows the men and women who pay for such services to engage in higher end work.