And yeah, there are issues with construction crew unions and no-bid contract corruption, but who's to say that also won't happen with a private monopoly?)
The private 'monopoly' would still have to provide good service, at a good price, or a competitor could undercut them. A natural monopoly is still subject to market forces, even when it's alone. The threat of competition is just as good as actual competition, most of the time.
The chief problem here is that land is the one monopoly you can't undercut. With data, you can switch to multiple carriers, including radio. With electric, you can switch to generating some of the power yourself. But if someone owns all the roads, there's really no way to compete, since there aren't any easy substitutes. I guess this may push more research into automated flying machines, or teleportation...