The idea of having Boring Company pay for the construction and then reap the profits strikes me as a really great idea. When the government pays for construction, the contractors don't have a strong incentive to actually finish the project. They'll keep getting paid no matter how long it runs over or how over-budget it goes. But in this situation, Boring Co won't make any money unless and until they get it up and run…
The reason you don't want to do that is because the private enterprise wants to maximize its profits. There is nothing wrong with that but in order to do so they will cut corners where ever possible and do the absolute minimum for upkeep.
Yes, this happens with poorly managed and financed public projects as well but public infrastructure can and sometimes must operate at a loss.
In fact a lot of public infrastructure does not pay for itself and is subsidized by tax dollars.