It's amazing how many people in America think our car-dominant culture is just the result of market forces.
The truth is that land use for transportation is zero sum enough to make market forces impossible to use in the way we expect for most consumer services. If you want choices, the best option is to support multi-modal transportation, which in most US cities means reallocating resources away from cars (who currently have almost all the resources) and towards walking, biking, and transit.
Free markets need very specific conditions to lead to the optimal solution. I don't think the transportation market fits the requirements.
[1] https://en.wikipedia.org/wiki/List_of_motor_vehicle_deaths_i...
[2] https://crashstats.nhtsa.dot.gov/Api/Public/ViewPublication/...
I think it would be dubious that private roads would require liability insurance. At present, liability insurance is only widespread because of regulation (i.e. a non-market mechanism). I do not see the economic case for the owner of the road to require liability insurance of its users.
Insurance wouldn't cover the full cost either. According to [2], total car insurance losses were $117.5 billion in 2011, compared to the $242.0 billion in economic losses cited in my previous post.
[1] https://www.ibisworld.com/industry/automobile-insurance.html
Well that's your opinion, at least I find the framing and connotations of that word problematic here, but I agree to let the market sort it out.