How about just enforcing the laws that are already on the books?
I believe you are correct to a point. I recall of being taught how many were turned back at Ellis Island for health reasons, wrong paper work, yes even wrong country at times, no sponsor and so on.
I served in the service with a few guys who were able to bring their families ( wife and children) into this country by signing up for military service, they were from Panama, Mexico, Peru and two from Africa, I don't recall which countries.
Even in my wife's' family their were a few turned back in the late 1930's for some legal reasons.
As a professor told us in class immigrations of the illegal type isn't a problem till they kill one of your family by accident or not, kick you out of your house or cost you more money. He said it is why we should watch our borders and allow the kind of people we need, it could be doctors, teachers, labors, and what ever else but it is our need. If you wish to sponsor a person and take responsibility of the person and family coming in, paying for them and helping them for four or five years they get on their feet, that would be fine too. But not just come in because you want to and not care for our requirements and our people.