How do we do the things you propose? Enforcement, ok, good, fine, but don't think for a minute many employers won't go around the system, they have been for years. We still have to deport. We have to start somewhere.
First give employers the tools to determine if workers are illegals, then fine employers for each illegal worker and require them to pay for all the social services used by their illegal employees and their families. Jail time for repeat offenders.
A few company owners and presidents bankrupt and serving time in jail will focus the minds of other employers on this issue.