I have to fill out mounds of paperwork if I receive any external income
I'm not sure exactly the definition of external income in my case, but I know external employment counts and I've also heard coworkers brush off 1-off side gigs because they'd have to fill out paperwork.
The companies recourse here is to terminate you for violating company policy, which could open them up to a wrongful termination suit for trying to enforce a policy that may or may not be legal in that state. The company suing the person in this case has no value nor any outcome where it is a net positive for the company, and the company knows that and they just hope you don't.
I work for a municipal service. It's conflict-of-interest compliance.
In short, it absolutely is their beeswax.