It starts with a widely perceived problem that has widespread support: that the country is full of illegal workers.
Whether or not this is actually a problem is a matter that can be debated, but that's not actually the relevant bit. The relevant bit is that it's popularly seen as such in many quarters.
The solution looks simple: create e-verify.
All employers are required to check citizenship/work eligibility when hiring.
It seems simple, straightforward, and people will support it because it does not harm them, it keeps others from competing with them. we're simply solving the problem of "stop flooding into work, there are no jobs for you any more." (Well, except in agriculture which is being carved out because no one is crazy enough to block that and let crops rot in fields or field lie fallow.)
But there is a much more subtle twist here: it's a classic trojan framing.
You lose the minute you wheel this thing in. You have already surrendered, you just don't know it yet. "There is this seemingly simple thing you want" is dangled, but what it's really doing is turning a right into a privilege and gathering the power to administer that privilege into the hands of an unelected, unaccountable government agency run by people you have never heard of and probably never will.
I will remind you of Coyote's Law: "before granting any power to government, first imagine that power wielded by the politician you hate most, because one day, it will be."