The problem you are running into is that your particular imaginary ungoverned world includes some features only available via government in the real one. It's like a sunny science fiction world without much gravity that has a thick atmosphere for flying people to breathe - it's fine if everyone knows it's magical fantasy, not so fine for designing real life asteroid habitats. Something has to hold the air in place.
When you marshal these fantasy worlds in support of the people with Lester Maddox's real world agenda, as in Indiana, it's not fine any more. Black people can't breathe.
Feel free to think that government is as necessary as gravity, but this is your personal religious belief and irrelevant for me.
That some libertarian ideas about freedoms may be harmful, if enforced by states as obligatory, is clear. This holds for the freedom of contract as well as for sexual freedom - if one of the free choices allowed in a free society becomes an obligation enforced by the state, we are not in a libertarian society, and the results may be harmful. But this harm caused by the state is not a problem of libertarian ideas.
And also with what you call private goods, such as education and health care and transportation and sewer and water and so forth and so on. Illustrating the uselessness of the distinction as you attempt it here.
You may think it is useless, but this is your private point of view. I follow here standard mainstream econonic theory that for private goods the market is sufficient, and gives optimal results. Feel free to argue that mainstream economic theory is wrong, I will even support you in some important parts (Keynesianism is almost completely wrong). But in this part mainstream economic theory is fine, and my only disagreement is that:
1. The role of the public part is in many question overestimated (education, health care, infrastructure).
2. The ability of morals to solve public good problem underestimated.
3. The inability of the state to solve public good problems is ignored.
The problem of (3) is that, on the one hand, the state is not restricted to the market solution. But he has, on the other hand, nothing to guide him about what is the optimal solution. Thus, the solution of the state may put far too much ressources into a particular public good.
It is a theoretical certainty that stable suboptimal equilibria exist in ideal free markets, and they are an observed reality in all kinds of industrial market economies regardless of their degree of "freedom". They are a property of market exchange in any sufficiently complex economy.
This requires, conceptually, that you know what is optimal. So, usually behind this claim of "suboptimality" is some ideological prejudice. Like the particular ideological prejudice in favour of one particular form of forced child labor - learning in a public school.
What else? Some examples of large investments, which make only sense as large investments, and there it may appear difficult to raise enough money for this on a free market. In most cases not plausible, given that free markets are able to collect quite large sums for particular projects.
No, it doesn't. Prisons provide significant and necessary infrastructural support for the creation of their markets, and I even listed a couple of these necessary features for your edification (various aspects of accountability in exchange, for example, are rigidly enforced - cheats can't leave town).
That some aspects of what the state is doing - as unwanted side effects - even support illegal markets, ok, this is indeed a quite typical consequence of the behaviour of states, that they usually do not take into account most side effects of their actions, and that these side effects are usually harmful.
By the way, the illegal markets exist not only in prisons, but the same illegal market exist outside too. Thus, the infrastructure "provided" by the state in support of these illegal markets is certainly not decisive. (You have a point that I should not have used the word "prove", because what requires a comparison of very different situation can be only a plausible argument and will never be a mathematical proof. But that's all. That the existence of markets even in prisons, where the state actively fights against these illegal markets, is a very strong argument that no support of the state is necessary nor for markets themself, nor for media of exchange.)
I'm not sure what you mean by "the will of the State", but I recommend you pay more attention to the deeds of the State if replying to arguments based on those deeds and their effects.
This would be a nice recommendation for statists. They usually think that it is sufficient to show that some market outcome is not always optimal, but completely ignore that they have no proofs that what the states provide is optimal, or even better than the market where the market is not optimal.
But fine that you, in this particular case, prove that you understand that it is not clear at all what is "the will of the state", which is certainly quite different from what politicians promise, and that the deeds of the state have also side effects.