We've tried doing that. It's failed.
You're wrong. There is little or no enforcement of the employer ID requirements or of the so-called "employer penalties." The big INS raids that used to take place 15 years ago don't happen anymore. That's all because the INS is not enforcing the law. They won't even let local law enforcement check or do anything else about criminal suspects' immigration status here in Southern California.
On the other hand, the EEOC has regulations prohibiting employers from "discriminating" on the basis of "immigration status." Know what? They enforce those regs.
Don't tell me the immigration law "doesn't work." It's not only not enforced, the Feds also have "civil rights" regulations in effect that provide legal disincentives, to add to the economic ones, for employers to try to obey the immigration laws. We don't have the will to enforce our existing immigration laws, let alone enforce new ones.
I think the employment laws must be strict enough to make employers check with the INS and SSA the validity of the documents. This is the way it's done in most states at DMV offices. Reliance on the INS alone won't work, because the INS may have some chance of success only when the violations aren't on so massive scale as now.