Well, I think around here, they ban "motorized" vehicles (except for wheelchairs and the like). Legally, I think e-bikes are classified as "non-motorized" vehicle.
As for Nashville, about 1/2 of that "Shelby bottoms" greenway was my family's land back in the day. Oh, the stories I've heard bout that area. Makes for a pretty bike ride, I'm glad its getting put to good use. :-)