For effective tracking you will need a national Id card for us nationals. That is something many different groups in the US are against for different reasons. Such an id could solve a lot of other problems and of course will create its own problems. But I think the net benefit outweighs the loss.
You could also potentially do it by retaining positive tracking of non-permanent visitors and then going after them as soon as status becomes disallowed. It would be challenging to locate someone in a disallowed visa status on transition without a national ID card, but there are probably ways.
One option might be to enroll them on entry with lots of biometrics, and then do a "not on blacklist" biometric check at government-provided services. That wouldn't necessarily require identifying the US people, just validating that the US people are not the other non-US people.
If we can dramatically reduce the number of illegal status people by making legal immigration reasonable, it becomes easier to go after illegal immigration with more resources per person, and higher penalties.
Other than going to the DMV once a few years, buying a house and flying, I rarely if ever use a government service that isn't transit. When I first come in I use a lot of government services, such as getting a bank account and an SSN, but that is a one time process that happens when your legal.
