Private government is really nothing new.
And I don't mean some arrangement between coercive government and "private" companies that make their money from government contracts.
That arrangement actually produces the inequality that people blame on free markets.
Notice how everyone is taxed. Yet the government spends the money enriching the government defense contractors, handing out subsidies to already rich oil companies, bailing out sub-par automakers and predatory bankers…
In contrast, private government is not coercive.
No one forces you to live in a neighborhood that charges a "tax" like a homeowners' association fee. But when a government monopolized all the land, you have no choice but to pay the taxes they demand.
And under the private model of paying for government-like services voluntarily, the money flows from the rich to the rest of society.
So coercive government is actually used as a tool to further enrich the protected classes intertwined with government. This isn't all rich people, it is just the ones who got rich from cozying up to politicians instead of providing goods or services that the public voluntarily pays for.
I'm actually writing this from a private community in Puerto Rico. There are roads, security, trash pickup, and even servers to bring you drinks on the beach.
The difference is that you pay for all of it, either a la carte, as with drinks on the beach, or through the HOA fees, as with the security and roads.
And I can't afford it. I'm just lucky to have a friend generous enough to let me stay for a week.
Most people can't afford to live in entirely private communities like this. And that is the main criticism people have for private government.