and when they have fake documentation, then what?
“and when they have fake documentation, then what?”
It shouldn’t matter; either you have an employment history in the database or you don’t. The system is designed to vet IDs by checking them against existing databases, primarily social security.
Of course, with fake documents it becomes something of an arms race. Once you screen out the lower quality fakes, you force the establishment of fraudulent data and better fake IDs with the data to back them up. But that costs money and the economic incentive begins to evaporate. At a certain point it becomes too expensive for the illegal alien to get the documentation they need to secure employment.
You’ll never completely get rid of people using fake documentation. But if you make it expensive enough, you can largely take care of the problem.