illegals dont have SSNs. EMployers have to run ssn verification before they hire you. These checks are instant and this will popup immediately. Can they still hire you - well maybe for janitorial -> manufacturing jobs I'd say highly unlikely.
Employers are supposed to have employees fill out an I-9 form that verifies employment eligibility, but the penalties for noncompliance are at most $1,100 per form -- basically slap on the wrist territory [2].
The irony here is that the anti-immigration wing of the Republican party is now in open conflict with the business wing of the party, which benefits heavily from hiring ineligible workers for low wages.
[1] https://en.wikipedia.org/wiki/E-Verify#Operations
[2] https://www.thebalance.com/i-9-form-employment-eligibility-v...
If you really doubt that illegals can work in manufacturing, try googling for 'immigration bust factory', or similar stories.
You're right - you do need a valid name/SSN/birthdate. It doesn't have to be yours. And if a company wants to be disreputable, all they need to do is keep running them until they hit jackpot.
Because then they can claim that the other people were, uhhh, fraudulent. And this person passed their Everify. It's not like there's any proof that the Everify was run 10 times prior, to find a good name/ssn/DoB.
Oh wait...