I would think that if a motorcycle can cruise thru National or State parks, a motorized scooter, bike, or most anything else could 'IF' it is a tagged/licensed and insured vehicle...
otherwise, it might be considered a 'toy', and not allowed on the roads, and probably since it has a motor, also not on the walkways, paths, or even bike paths.
These same questions come up many times with golf carts, atvs, and ohrvs, in many jurisdictions and government regulated parks/areas.
Some small towns, especially in golf communities, allow golf carts on the roads IF they reach the minimum level of safety - such as turn signals, brake lights, and also include tag/license plates and is insured... with a licensed driver at the wheel.
I imagine that motorized bikes are in a similar standing in many areas... they can't attain a minimum speed on major roads and interstates, and they don't carry the aforementioned standard safety equipment, such as signals and brake lights, yet they are considered a motorized vehicle, as far as the authorities are concerned, putting them into the same 'laws' and restrictions of roadworthy vehicles.
Of course, if you 'look' like you're peddling your bike, maybe nobody would be the wiser!
'14 Palazzo 33.3 bunkhouse 34' diesel
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100,000+mi since '14 - US, Alaska, Canada.... formerly Georgia, then NorthCarolina, now Alabama : )