This is something I hadn't thought about. I'll have to consider it.
The other part of the problem that isn't being addressed is the magnet that the work here creates. There are some jobs that migrants are doing that Americans just don't want to do. It seems to me if we could create a system for temporary workers that would carry punishments for employers as well as the workers that are so severe it would create an incentive for employers to work within the system.
If the lure of jobs is controlled and benefits are denied to those outside the system the magnet ends.
Now the problem of the children brought here as babies has to be dealt with. A large group of children raised here with no ties to any other country live here illegally. They can not get SS#'s and are in a permanent underclass. This problem will just get worse.
If you buy produce in a grocery store, probably most of it was picked by an illegal alien.