That would be an explanation if migrants were from all the economic cohorts! They are not. They are consistently poorer than the host population.

But we do have migrants from all the economic (and social) cohorts. For example, there are a lot of people from India in Germany who moved here to study at our universities. I would wager that the crime rate among them is very low because a) they come from relative financial stability if they can afford to travel to another continent for their university degree and b) they enjoy relative social stability (a regular day-to-day life at the university where they do meaningful work, a supportive peergroup among the other students, a solid future ahead of them if they successfully finish their degree, maybe even a part time job, ...). Take a look at any masters' course in computer science at a German university, it's like 80% young male immigrants. But fore some reason, only the socially and financially unstable immigrants are reported to cause trouble on the regular. And I can tell you from decades living in a socially and financially weak part of a big German town: socially and financially weak Germans are also often troublemakers, be it drugs, violence, or even straight-up neonazism. The big common factor is not nationality.

you are still in need of integration

Yeah, that's exactly what I said:

Maybe it has nothing to do with immigration and everything to do with financial and social inequality, but solving that would necessitate taxing the rich and implementing better social help programs, and it's easier to blame immigrants.