If they genuinely can't work out what you owe, why bother paying it at all? Shouldn't there be a massive tax evasion problem?

There is actually a pretty massive tax evasion problem. Or at least the IRS is pretty sure there is, but they don’t have the resources to go after even a small fraction of them. The only thing that keeps people honest is the worry that if they lie, the IRS might already know (based on e.g. 1099 reports that go to the IRS), or they’ll get audited (which actually happens very infrequently).