I imagine that the majority of private landlords are not ruthless exploiting business people - far more typical would be people who have bought one or two properties as part of a pension plan, or parents who have bought a flat for a student son or daughter and find themselves having to check up on their child's housemates.
As Glamma says, such people will not have the resources to verify whether documentation is genuine or not. There is already a problem for college authorities having to check the bona fides of overseas students, and if it is difficult for them, how much more difficult will it be for small landlords.
I have no problem with landlords having to comply with stringent health and safety requirements, because it is their responsibility to ensure that their property is safe to live in.
But they are not responsible for the government's immigration policy, and they should not be criminally liable for the failures of a government agency.