The idea that one could completely do without public institutions reflects a departure from the "back to the founders" idea that many identified with libertarianism. The founders don't seem to be anti-statist enough for some libertarians. The problem is that societies with no sense of the public may find it hard to defend themselves against foreign or domestic enemies. They may also find it difficult to rise above the rivalries of clans, factions, or tribes to the kind of impartial sense of rights, duties and obligations that characterize freedom and modernity.
With regard to the specific question you address, it seems to have a lot to do with the modern concept of the public square and with contemporary patterns of migration. Prior to the 1960s a public square would likely contain a war memorial or other monument. It might well have contained a temporary creche or candle display sponsored by some non-governmental institution. It would probably would not have contained individual artists' self-expressions. This is I think something new. The older idea was that such sculptures were representative of the community's ideals.
If the idea of "the community" having ideas of its own has disappeared, one reason is that affluence has made us more individualistic. Another is that totalitarianism has made us more distrustful of such singlemindedness and more determined to assert that individualism.
The third reason is that recent immigration has done away with the idea that we could have any single "culture" -- though technology, media, and economics do a lot to make us more uniform. An earlier America or an earlier West could proclaim the naked public square because it was presumed that society was Christian anyway. In a more "ethnically diverse" land, rivalries between groups do tend to become disputes over the public square.
Libertarian societies tend to be successful at attracting people from many different cultural backgrounds. The problem is that once groups reach a certain critical mass, they want society to reflect their own group norms, or at least give more weight to them. This is the downfall of libertarian polities, and one saw such developments in Britain, Austria-Hungary and elsewhere at the beginning of the last century, after decades of relatively liberal (in the classical sense of the word) economic and political policies.
BTW, William Ebenstein's classic introductory work Today's Isms has been revised and reprinted by his son with a chapter on libertarianism.
The word public has a number of senses. First is public in the sense of "open to just about everyone." Second is public in the sense of state-owned. Third is public in the sense of common property. It seems to me you're conflating these meanings. Just because a property is privately owned does not exclude it from being public in the first sense.
Also the notion of law as a creature of tradition rather than a product of legislation seems foreign to you. This is unsurprising given the gutting of common law in recent decades. It is difficult for me to see how criminal justice requires property that is public in the second sense (state-owned) in order to function. Do we really require politicians in order to have law?
And what exactly do you mean by a double burden? Are you saying that you think a person could not purchase police services without the existence of state-owned property?! This seems so odd as to verge on non-sequitur. Perhaps I'm misunderstanding your point.
Entelechy in 18: one has to wonder what the benefit of a commons is.
My contention is that if the commons is somehow prevented from happening (e.g. forbidden by the constitution), a private commons will be created to fill the need for unsolicited exchange; the owner of such private commons will have to resolve cultural disputes based on whether they actually disrupt, and not based on "because-I-say-so".