If we strip your comment from the emotional affect, you're essentially saying that immigrants are happy to do menial jobs that the locals do not.

How do countries without large immigrant (I won't even go to refugees) populations handle this? Switzerland seems to be doing fine. Japan isn't falling apart because there's nobody to cook. Most of Asia is doing fine without allowing any immigrants.

So on a strict analysis of whether immigrants are beneficial to the host population, your argument isn't made.