Is Cooper's Hill Lane actually private for cars? There's a dead-end sign ("except cycles") where I placed the marker, but I've driven down the end of the road without issue and saw no restrictions. (approx where the driveway is, there are bollards preventing vehicular access, so I'd assume that's where car=no happens).

Certainly requires a 4x4 in wet conditions. Should also probably be marked, I almost got bogged down in 2WD mode in my car coming back up and had to switch to 4WD.