One of the other problems with the split labor market that mass illegal immigration creates is that it means that welfare state policy is put between a rock and a hard place. If welfare benefits are made more generous than the market wage for illegal immigrant labor, but only for native workers, then native workers leave the workforce en masse, with all the social problems that creates- and much of the native population can easily be disempowered politically now that they are no longer necessary economically. If welfare benefits are made sufficiently less generous, then native and illegal immigrants are competing directly for low-skilled jobs, and the market wage is pushed still further down. Here’s a simple (if messy) sketch of how that might work, with two different reservation wages (A- minimal welfare and B- Generous welfare) determining if the market wage is set by immigrant or immigrant and native labor supply combined.
If you think, as I do, that increasing automation is, sooner or later, going to displace a lot of human labor and require some degree of increased cash support if people aren’t going to become indigent, then this issue is going to become more salient rather than less. The sensible thing is, I think, to consolidate the labor force as much as possible, such that the same regulations and welfare eligibility rules apply to all. This could probably be done without much deportation, if the government were more enthusiastic about sanctioning employers who violated labor and immigration law. But I’m skeptical that that will take place.