Hey BU... I would add Landlords to your plan. If someone who should not be here cannot find work or a place to live, they will go home where they belong on their own.
The penalties would have to be harsh for the employers and landlords, but it’s really a very simple plan.
There is no way to check on citizenship for a tenant. I am in the industry and I am familiar with the requirements. Now, as for illegal apartments or renting over capacity, that is an entirely different situation.