That only works for 1 or 2 generations, when the immigrants' descendants find out what the pre-existing citizenry already has found out: that it's too expensive and difficult to have children, particularly when you're overworked all the time.
Plus, immigrants are usually only brought in for jobs that are shittier, harder, lower-paying, and with fewer benefits than the current population of native born citizens already have.
I'm a children of immigrants in the USA who is childless (hell, most of my cousins are too) and I live around immigrants of every nationality, and most either want to wait until 30 to have kids (and generally only like 1 kid at that) or want to have no kids.
It's like that in a lot of western nations with birthrate problems. It doesn't take long for new people to start acting like the old people if the conditions are bad enough, and they are.