As regards "private roads", it is the rare accident indeed that damages the road. Normally the road is just like, "SWEEP UP THE GLASS AND HOSE ME DOWN!". If the private road owner was so concerned about damage and liability, then they would buy insurance for this purpose and raise prices accordingly. Why would they want the paperwork nightmare of dealing with tens of thousands customers? If the damage is specific, they go after one or a few parties and if those parties can't pay up, then they will have their own insurance or contingency plans in place (like self insurance).
Whenever anybody says, "you need to have insurance" or "your contractor has to have insurance", then it is a damn good bet what they really mean is, "we don't properly mitigate risk ourselves, so we need you to do it for us".
I understand the lawsuit happy world in which we reside. That said, the motivation that mandates insurance is not risk mitigation, it is corporatism and foolhardy notions regarding the nature of risk and the nature of insurance. Mandating insurance is the first step towards "universal mandatory umbrella insurance". After all, you didn't create the universe, why should you be responsible for anything?