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In some(most/all?) states, all accidents over a certain dollar amount or that caused injury to any party are required to be reported to the police.

But the lack of police reports is why the article mentions that insurance companies believe the accident rate is higher than the 'official' number.

Anecdotally, the number of times that someone has hit my car, and then asked me not to report it in exchange for cash is non-zero.

Not necessarily. Major accidents will have the police involved, small fender benders might be handled with just an exchange of insurance information. If there's substantial damage though, the police will probably get involved since a police report makes collecting on insurance easier.

It's not uncommon to contact the police so that a quick statement can be made and the paperwork logged. It seems to help with the insurance stuff and getting a third-party's eye on the scene.