It only restricts the lawful.
Nope. Not if you set up the legal burdens correctly. The problem with the current system is that employers have the easy "out" from legal liability by claiming they relied on forged documents. Close that loophole with a better I.D., limit by statute the appeal process, and you can greatly increase enforcement.
The trick, though, is that you've got to give them an incentive to signup for the I.D.'s in the first place.