My family came to the US via Ellis Island. Compared to what people have to do today, their legal path to US citizenship was relatively easy. I see no reason that becoming a citizen today should be any more difficult today than it was in the early 20th century. Open a 21st century equivalent of Ellis Island, and let people become citizens.
The growth of a welfare system seems like the major change. How does that plan interact with the welfare system? If someone is impoverished in Asia can they get a plane ticket to the US and expect to eventually be entitled to a state-sponsored minimum standard of living? Maybe healthcare if the left's plans for that get through eventually?
1. You were free of contagious medical diseases
2. You were not in danger of "becoming a public charge" (welfare)
That plan is perfectly compatible with your concerns.