Open immigration worked fine up until the point that the government started handing out checks to people who hadn't earned them.
Third world people will come here by the 100s of millions if we let them, and do it without welfare or social programs.
If you live on a dirt floor struggling to live and to eat, and to treat illness, then you would prefer to do that in San Diego, or Anchorage, or Honolulu, than in the backwaters of third world countries, even if there was no welfare or government checks.