"Historically, in this country, the social contract has not amounted to "papers, please!" That was supposed to be the kind of thing reserved for commies and fascists,"
Popular myth and Hollywood. And yes, the social contract HAS generally demanded that you produce papers. You want a job? Papers please. You want a loan? Papers please. You want government benefits? Papers please!
"Also, driver's license does not actually signify either citizenship or legal status. The amount of supporting documentation that is required to issue one varies from state to state, and not all of them ask for an SSN. Of course, even if it did, not everyone has a driver's license - as the name makes evident, it's a document that is issued for a specific reason, and not all people even need it."
You're right that it doesn't signify citizenship or legal status, but it does establish identity. And the supporting documentation required to get one does require establishing said identity - or did until some of these states changed the laws so that illegal immigrants could get a driver's license (and those driver's licenses are different from 'normal' driver's licenses.)
"how exactly you imagine checking for it."
The same way we've already been doing it: you want a job? Papers please. You want a loan? Papers please. You want government benefits? Papers please! The problem is right now we're not enforcing it hard enough - _punish_ companies/people using illegal labor and they'll stop doing it once it's no longer cost-effective.