If roads were built by the private sector they would be well maintained and safer and less costly.
I'm all in favor of free markets - under a framework of sensible law that makes an effort to neither favor anyone, nor limit unnecessarily.
But the economics of roads are such that I'm very doubtful that having them managed privately would be good for anyone but the private road owners.
If the cost to maintain the road efficiently is X, then the government is likely to be inefficient, and will spend K*X. This tends to be because the road maintenance contract doesn't go to the company that would do the best job for the price, but to the politically connected company that will intentionally build the roads shoddily, so they can keep repairing them and charging for it. I've heard anecdotes where the company with the road maintenance contract will intentionally wreck a well built road that would have required little maintenance, just so they can repair it in a way that will require lots.
But the value to you of using the road is Y, and I'll bet this number is much larger than both X and K*X. If the private company can charge whatever it wants, it's going to charge you 90% of Y, which probably amounts to a factor of magnitude more than K*X.
To leverage markets to lower the price provided by private road owners, you'd have to have competition; but this would require competing roads between each pair of places. Barriers to entry are enormous; it is highly expensive and complex to acquire the land, even before the cost of building. Then there's the problem of permits, and how to ensure the government's neutrality. You can't not have permits, or road builders will wreck the land; but when you're requiring permits, the government is getting to choose which private company wins, and that's no competition.
Ultimately, there cannot be more than a handful of road operators connecting each pair of places. The barriers to entry are huge, so the few road operators will be able to reach a friendly agreement to maximize the value they extract, at everyone else's expense. What you'll end up getting is service comparable to your cable company - the most hated businesses in the US, for mostly these reasons.