"Give me your tired, your poor, Your huddled masses yearning to breathe free, The wretched refuse of your teeming shore. Send these, the homeless, tempest-tost to me, I lift my lamp beside the golden door!"

America was built on immigration from undesirables (an Irish or Italian Catholic ~100 years ago was treated as suspiciously as a Muslim is now), and it seemed to turn out okay. The key difference back then was there was minimal welfare, so immigrants had to find something productive to do or else they'd starve.

>At best — at best — the 1st world would rapidly equalize economically and politically with the rest of the world. And what would that accomplish?

This is already happening via outsourcing: see for instance the rise of China, hundreds of millions of people lifted out of poverty while working-class Americans get poorer, and China's rising political influence on other countries. Open immigration would encourage more competition between countries: a hellhole like North Korea would have to clean up its act if its citizens were free to get up and leave.

>And after such a grand experiment, the refugees would be hardly any better off than they are now.

Do you think current (legal) refugees would agree that they're hardly any better off than they were back in their original country? Why would illegal refugees be any different? (Often the only difference between an illegal and a legal refugee/immigrant is that the legal one is at the front of some queue, the illegal one is at the back).