It didn't used to be that way, and back in the 90s "Papers Please" was a reference to the German Gestapo.

ID to drive, ID to buy cigarettes or alcohol, ID to enter a bar has basically changed all of that. It has become de facto required to carry papers at all times, and show them on demand. Nobody makes the "Papers Please" Gestapo reference anymore.

Like I said before, I think Tennessee has the right idea, make it illegal to rent to illegals. The landlord is already vetting the person, so he can ask them to establish their legality and take the time to look over it. Bust a few of the most egregiously non-complaint landlords and let the rest of them do the vetting.