It’s the feds job to see that people are here legally, not 8 zillion employers. If you asked retailers to verify citizenship to make purchases, that would be effective, too, but again imposes a burden unnecessarily.
The idea IS to make it hard on employers. Get it? The harder the better.
I won't argue that point, but it's also the feds' job to enforce fair trade practices. An honest employer who insists on hiring legal employees only, is at a disadvantage bidding against an employer who hires under the table.
My employer used e-verify to confirm citizenship and/or legal residency. It is the employer responsibility. I have no idea if the people who work for my gardener are here legally.