Then set up stations at the border. All employers, and all illegals would have 30 days to check in. Businesses would have to request workers and show proof that the job had been offered to Americans. "Guests would be required to show proof of insurance and they and family members accompanied would be required to have a criminal and health check.
The "guests" would be required to be productive for America and not be eligible for benefits, including SS. A "Guest worker tax" equivalent to SS amount would be deducted. This would vary, dependent on how many children would require public school education.
Limit their work visit to two years at a time. Under no circumstances would I give them an edge on citizenship. One would have nothing to do with the other.
Finally,I would require registration but exempt the elderly already on SS . I don't want to be a nation that pushes the old people in wheelchairs to the middle of the bridge and let's them roll home.
But, in two years time, they can have created a home, had a child that is now an American citizen. Wonder how you solve the problem of sending them back to no home, no job without the same guilt/hardship problem we have with the illegals here for years?