I don't think the point is to eliminate all illegal work.

But the new form of ID makes work place checks real easy and fast.

Add a real hefty fine for the owner and possibly ban from conducting any form business for a few years, that will have undoubtedly have effect.

There are already hefty fines for owners of businesses where the people are not working "legally". There's a "share code" that employers are supposed to check to verify visas. All the laws and machinery is already there and it does not look particularly high-cost to me.