I think you're probably too cynical, though it could certainly get as bad as you suggest. The problem is addressable even now -- a reasonably-effective guest worker program would probably do the trick: ensure that Americans hire only registered "guests" (which can be done via internet!); and make sure that the "guests" are taxed, through witholding.
Taxes will address many of the imposed costs; registration and tracking addresses the national security issues; and registration and witholding requirements address the financial incentives for businesses to hire illegals or guests.
But supposing you're right, and the problem cannot be fixed: wouldn't it make more sense to make the best of it?