It's not really a government as you can't (ideally) force people to participate even if they don't want to (or if you do it's through tamer methods like boycotts or whatever rather than threat of imprisonment.) And it doesn't have to be a central organization. It sounds like a minor difference but it's really not.
Importantly most things that are done by government would be done completely differently and by private organizations. The point is, that if the way government does something is truly superior, then people would still do it that way. They just couldn't force people to participate or pay for it. But you could still have a non-profit with elected officials and all that, if it's really better and can compete on a free market.
Again, I'm no longer certain this is the best system as there really are things where a free market doesn't lead to the best outcome and participation has to be forced (like public goods.) But there are ways of funding public goods without a government that might work, and most of what government currently does doesn't fit in that category anyways.