Your homeowner's association is different from a government entity, how again? If it were the county maintaining roads, the county would collect taxes and maintain the roads. Since it is a neighborhood, the association collects dues and maintains the roads. It is the same thing on a smaller scale. Perhaps the smaller scale makes it more efficient, and perhaps not, but it is really no different in kind, just in scale.
As for roads that benefit the general public, rather than just a small group of homeowners, a private entity building and maintaining roads would have to make a profit somewhere, or at least meet their expenses. To do so, what would they do? Collect tolls? If it were one limited access highway, maybe that would work. Were it a network of county or state roads, it would require a huge number of toll booths.
It is far more practical to collect a gas tax and use the money to build and maintain roads.
As for your rich people who won't allow the government to buy their property, that's just another example of the undue influence of money in government. You're no doubt right that, were it a middle class subdivision, the highway would already be built and the property taken via imminent domain.
So, how would a private entity manage to get the property owned by the wealthy homeowners you mentioned?
