And that notion is what many of us reject.
To argue otherwise requires a fundamental rejection of the legitimacy of the state
Not exactly, although plenty of Libertarians do reject the legitimacy of the State out of hand. I find the issue to be more nuanced than that. If you hold, as Bastiat did, that "the law" (aka "the State") can be something like - but no more than - the "collective extension to our individual right to self defense" and that no mere aggregation of individuals can have a right to do anything that a single individual doesn't have a right to do, then you can conceptualize a State that does things like, say, protecting private property, but which can't lawfully require payment of taxes.
All of that said, I didn't come here to write pages and pages on political theory. There are entire books that treat this entire topic much better than I could. I'd refer anybody who is interested in this to The Law [2]by Bastiat, as well as the works of Lysander Spooner[2] among others.
You can only do that for the specific case if you believe that either (1) the specific concrete details of property rights are matters of self-evident and immutable natural law, not contextual judgements that involve a balancing of interests, or (2) each individual is empowered to define and impose their version of property rights on others. Otherwise, a state with no power that isn't exactly that held by individuals comprising the state cannot define property laws, much less enforce them. (Technically, it can't define property laws in the first case, either, but it can recognize the concrete and specific elements of, and enforce, the laws that were carved on stone tablets by the invisible hand.)
Also, Lysander Spooner literally is against private property. He was part of the Socialist Internationale. The validity of his rejection of the state on basis of unjustified hierarchy also led to his rejection of capitalism on grounds of unjustified hierarchy. Instead, he was a mutualist. You know, the political economy for which one of it's foundational texts is What is property, which coined the slogan : "La propriété, c'est le vol", by Proudhon.
Bastiat attempted to use this essentially anarchist analysis of the state, but exempt private property from it by inventing some weird standard of government action that has no deeper justification and is pretty much axiomatic.
This is a huge simplification, but if you want more detail you can read "Why Anarcho-capitalism is not Anarchism" either on the Anarchist Library or, if you use Debian, via apt-get install anarchism.
As for Spooner: I am not saying I agree with his analysis on everything (same for Bastiat as far as that goes). I'm just pointing out that those two represent some useful sources of thought on the illegitimacy of the State. That was intended mainly for people who weren't already involved in studying this particular issue.
The question of "what is property" is an interesting and fundamental one for sure. I'm still chewing on that, honestly. Some days I think I know, other days I'm convinced I was wrong before. Lather, rinse, repeat.
If you have to axiomatically state that private property is good and government is bad, then I don't really see why anyone would have to follow your arguments, because not only are such axioms ridiculous, as seen by the attempt to create a rationalization for it. If the argument according to which the state is illegitimate has as a consequence that so is private property, but you still axiomatically state that private property is legitimate, you've also invalidated the original argument that the state is illegitimate, and now your entire worldview is based on two absurd axioms, and not any logic.
A moral philosophy or political economy is only ever valid and worthwhile if it is based on axioms/values that the rest of society share. For example - negative freedom is good, positive freedom is good, prosperity is good, and so on, which you'll find that people tend to agree with inherently. I don't think that there's anyone for which "private property is good" and "the state is bad" is an axiom.
Just to be clear, we don't have any problem with schools or libraries. We just think the model of having governments collect taxes and then pay for those things is broken. In consequentialist terms, it's broken because it removes competition from the equation, and competition generally leads to progressively improving products and services, and/or lower cost goods and services. In deontological terms, it's broken because taxation is theft.