If we require instantly verifiable proof of citizenship or legal residence prior to employment, school enrollment and regular well healthcare, many if not most who cannot prove legal residence will go home. They'll have no choice.
The argument "you can't deport them all" is nonsense. No one's asking to.
By the way - - when I go on a business trip, I need to produce my driver's license at the airline counter, at the baggage drop off, at the gate, at the car rental counter, when exiting the car rental lot, and at the hotel check in desk. I don't feel racially profiled at all.