This simply isn't true. They're not going to be paying for housing and services and having a job if they commit crimes and go to prison. Certain host countries have found that certain immigrant groups tend to have very high crime rates and cause a lot of social problems. Extremely low-paid immigrants, even if they have a job, aren't going to be paying much in taxes under a highly progressive taxation scheme.

High-skill immigrants are almost always a big win for the host country because of the things you cite, and more (they bring innovation). Low-skill/low-education ones are a crapshoot. It's better to have an inverted population pyramid than deal with all the social ills that come from immigrants that cause too many problems.