I agree that LP is not subversive in any trasonous or otherwise negative way, sorry to have made that impression.
The point is that country-state (indeed a better term) is what we know. Perhaps the system can be transformed legally along the minarchist or anarchist lines, but it will be a different system once it is accomplished.
I understand that we have private security and agree that it may provide a kernel of law enforcement independent from the state. But we don't have a system of law separate from the state, other than completely deprecated common (customary) law. The medieval system was not merely hierarchical, it had independent subsystems of justice. The church law operated differently and independently across local jurisdictions. The king's court, ditto, and the local manor court was also independent. The church, for example, could not dictate the resolution of property disputes, and the king could not dictate what heresies were. When the king attempted to defy the Church, or the Church tried to dethrone the king, it was a crisis, resolved through diplomacy or arms. Therefore, these systems were truly independent, and the Church was not a super state, and it did not "run" any states. It was something similar to the system of universal rights that the libertarians hope to implement (do they?).
If I miscontrued any of your points, I apologize. I do not intend to obfuscate. Yours are long posts, and I cannot pretend to answer everything in them with precision. I prefer to make my point of view clear and let the reader decide.
The medieval system was not merely hierarchical, it had independent subsystems of justice.
On this we disagree historically speaking, in as much as I say there was no medieval system of justice that can be particularly described as such.
I also do not agree that the church was not a super state. Using the argument of limited jurisdiction to say it was not a superstate, overlooks the fact that all governments throughout history have exercised limited jurisdiction to some degree or another. And so it was with the Church.
Your idea that "the Church... did not 'run' any states," is also not historically correct as you state it. Church lands, were quite often a state unto themselves, sometimes subordinate to other non church states, and sometimes completely independent of all other non church states.
Circumstantially, I can agree that the church you speak of "...was something similar to the system of universal rights that the libertarians hope to implement." Or at least many of them hope to implement. But such a notion at this time does not best describe libertarianism, as the considerations and options are far to numerous. It also implies a settled issue, that the libertarians have not even come close to settling.