Nah, this is a filtering criteria. This is pay-to-illegally-immigrate. Republicans are fine with over staying visas or that kind of illegal immigration, half their spouses do it. Iirc milania did it.
This isn't true. Republicans love illegal immigration because you can underpay people and avoid a lot of regulation.
It is trivial to implement e-verify for employment for everyone. It would nearly eliminate illegal workers. If Republicans (or anyone) cared about stopping the hiring of illegal workers, it is a trivial process to implement. No one wants to do it because then Americans would unionize.
They only want rich criminals. Not poor ones.