You're right, of course.
Frankly, it's amazing to me that I can check my bank balance, instantly, from anywhere in the world...but I can't easily resolve whether or not someone is a citizen of the country.
If you'll indulge me for a sec.... a few years ago my company inherited an employee from a takeover. The guy was (Bosnian? Croatian? don't remember...), his immigration status was pretty murky, and he was working hard to keep it that way.
It took my company the better part of a year of jumping through various hoops to determine that his eligibility to work had run out. I don't know all of the particulars, but I know that the HR guy was pretty diligent about trying to figure out this guy's status.
Sez me, here is a company that is TRYING to obey the law, working with a single person and getting a massive, nearly year-long runaround from various government constituencies. How difficult is it for a factory, say in New Mexico, with 2000 workers, half named 'Rodriguez', and all with "murky employment status"?
and the other half named Martinez