However, today's LEGAL immigration, at around 1,000,000 per year, is about the same, as a proportion of the total population, as the "restrictionist" period 1925-29.
Personally, I think we do need many immigrants, but we should make them bid for the right (in addition to whatever background, health, etc checks we want). Brain surgeons and opera stars would bid high. People who want to hire "cheap" gardeners would have to pay a stiff fee for the privilege. Either set a fee level, and see how many want to come at that level -- or set a number, and see what price clears the market. In either case, enforce the law, collect back taxes, and clear out those who won't comply.
I think it is not an accident that the major restrictions were implemented after the first World War disrupted the migrations from eastern and southern Europe, forcing employers who had gotten in the habit of using cheap immigrant labor to either mechanize or hire Americans (often blacks).