You are quite correct. There are laws employers must follow including filing I-9’s. E-verify, a free system, makes it much easier although I-9s are still required. AZ mandated that all business must use E-Verify and SCOTUS upheld that mandate. We must make E-verify mandatory nationally. It is currently voluntary.
Certainly a government that wants to maintain databases on citizens should be keeping one on foreign nationals within the borders. I agree, E-Verify should be mandated nationally.