Perhaps I should have said: Once an immigrant accepts US citizenship he is no longer welcome in his country of origin. A friend of mine was unemployed here in the US while he had a job opportunity in his country of origin in Europe, He did not get a work permit nor a residence visa from the country of birth.
That’s true, and it at least sometimes works both ways. To get even a tourist Visa to the US, from a 3rd world country, one must have a good job there, references, a US “sponsor”, in some cases be able to prove things like ownership of residence / property, money in the bank, etc.
Of course, a problem with some of these countries is that such proofs are not hard to buy, then just find or buy a “sponsor”.