Yes. $1,000 fines per day per employee on a company employing an illegal would cut the magnet. The Feds know when an employee is using a stolen SS number because payroll credits come in from multiple locations, for more work hours in a week than exist in total. Each person using the same SS number needs to be identified, the American protected, and the impostors ejected.
Prior to about 2009 or so I remember reading about employers who were drawing prison time for knowingly hiring illegals.
I wonder what happened after that?