People fail to understand how hamstrung employers actually are. Sure, many shady small employers want to be able to employee illegals to save money, but legitimate employers want nothing to do with it. There are criminal penalties available and no executive with a legitimate employer wants to be jailed.
There is a very sophisticated network of operatives giving illegals fraudulent work authorization documents. Employers are required to accept at face value those documents which appear to be authentic. So you have likely millions working in the US with fraudulent documents where the employer has no reason to believe the workers are not legal.
I represented an employer that had a large facility in a tiny midwestern town - they received information that called into question the authenticity of the EADs of many employees, so the employer did a complete re-verification of the workforce (a very risky move under the law). Of the roughly 850 employees, half just simply disappeared rather than face renewed scrutiny of the EADs.