I see it as keeping to domains of dara appropriately searate. The employer needs legal status information on only the prospective employee. The government needs legal status for all, but does not need the information of who is applying for what job.
I do not know of a better solution to the issue.
“I do not know of a better solution to the issue.”
Investigate and indict some high profile fedgov officials who are suspected of aiding and abetting illegals and illegal entry.
A witch hunt similar to the Russia “investigation”, if you will.
You’ll see the whole problem disappear overnight.