Sure, in 60 years or so.
I'm not saying the authorities should have to come down on these companies like a ton of bricks all at once.
I am. I'm saying that our program should be that if employers can't prove that all of their employees are legal, or have sponsored guest application status within a reasonable time frame, like 6 months or even a year, they should be heavily fined or prosecuted. That's the only way we'll end the problem.
So you claim. A simple way to test it is to pass the legislation and see what happens. It won't make anything worse, and if things turn out as glacial as you predict, then you'd have some ammunition to use to push your plan. And the groundwork would already be laid for it.
There can only be one logical reason for being so adamantly opposed to it as many Beltway politicians are, and that is that they're afraid it will work, not that it won't.