I'll go a step further; I advocate a thirty year timeout on all immigration, legal and illegal.
Our culture is now under great stress and one of the primary reasons is that, over the last thirty years, we have taken in many millions of immigrants both legally and illegally. Under the banner of multiculturalism, an absolutely disastrous idea, many of these newcomers, and perhaps most, have seen no particular reason to become Americans, and so have not. By becoming an American, I'm referring to the adoption of American culture as one's own.
It will take at least thirty years more to assimilate the people who are already here.
I agree with your sentiment, but there is no way we can cut off immigration completely. If we go back to the 195,000 a year during the period 1921 to 1970, we can achieve what you want.