I'm curious to know what countries you're talking about considering that the United States has some of the least restrictive immigration policies on Earth. Furthermore, east Asian nations (Japan, South Korea, and Taiwan) would contradict your claims as they've got highly restrictive policies and enjoy extremely low unemployment. Hell, it was less than 15 years ago that Taiwan even allowed foreigners to sign up with a mobile phone plan under their own name. Japan, with a long stagnant economy, is at 3.8% unemployment, Taiwan is at 4% and South Korea is at about 3%.
There are significant benefits to immigration, but those are felt in the long term. Immigrants offer a diversity of ideas and talent for the host country. That, however, doesn't mean that every random individual who tries to enter this country deserves to be here.