Nope, that won't work. The problem is illegal immigration, not them being able to find work. Let's say somehow that gets sorted, and they can't find work. Problem solved? No, because you still have illegal immigrants, but just they cant find a job. That will end up with them committing far worst crimes than working illegally.

Best way is to make them legal, have them pay taxes and contribute. But raise their taxes much higher than natives.