> Indeed the main issue is that there are dodgy employers and landlords who simply flunk the law

I'm surprised that this is your framing, I don't think I would hesitate to offer a willing tenant or otherwise good employee a job because of their legal status. Mostly just on principle, it's not my job to be an arm of the state and I resent being deputized. They're physically here in my town, better they have somewhere to live and a means of supporting themselves rather than being homeless. If the state wants to find and deport them they can do it themselves on their own time.