I'm involved in hiring sometimes, and I know that the requirement is for a job applicant to provide various proofs that they are legally eligible to work in the U.S. As far as I know, my company stays in compliance with that law.
But your list provides an interesting angle on this issue. Assuming your list is correct, then there are plenty of major, big-time companies (not just the small-time farmer who is trying to get his mushroom crop in), who are in blatant violation of the law. If you can find this out, then the government can find this out. If the government can find this out, and these companies continue to hire illegals, then it means that the goverment is failing to or is refusing to enforce its own laws.
Given the trio of millions of illegals on the ground, big-time companies that are hiring them, and a goverment that won't enforce its laws, what real hope is there that this problem is ever going to be solved?
None. The powers that be want the cheap labor. And we'll bend over and take it as usual.