There is no controlling illegal aliens if aliens in the country are not controlled. That includes preventing them from working here. If an "American" business considers that "excessive" too bad. Unlike all the other unnecessary/questionable regulations, national sovereignty is the first responsibility of the federal government under even the most traditional conservative interpretation of the Constitution.
So just go after the illegals. Directly. Go after anyone with a temporary visa. But don’t go after all the businesses. You said “preventing them from working here.” I would prevent illegals from coming and staying here no matter what they do and leave everyone else alone.
What about all the illegals that don’t work? I would think that’s the biggest problem because they are causing more crime, taking money and not producing anything. I don’t understand why you’d favor this roundabout system with layers of bureaucracy and punish everyone EXCEPT the crooks and freeloaders.