Er, this is already the law right? Every job I’ve held in the US, I needed to prove I’m eligible to legally work. Obviously this doesn’t work to resolve the issue at all.
Illegal immigrants have been working small restaurants, ethnic supermarkets etc. since time immemorial, law enforcement has turned a blind eye is all. Now with the rise of gig economy, there are tons of jobs where you never even see your employer, so it’s even easier: there are citizens/permanent residents/others with work permits renting out their gig accounts like Uber Eats to illegals.