Subsidizing the use of roads by letting people use them for free after spending a great deal of money on building and maintaining them is not "market-oriented". Charging people to use them is the closest you can get to market-oriented if you are already stuck with the roads. The most market-oriented solution would be not to build public roads in the first place, but it's too late for that.

I suppose we could just demolish all the public roads and start over.

Urban roads are a natural monopoly, you're not gonna build multiple parallel sets of roads going along each block.