In your post, you refer to "shared natural resources" and then "deprive others of access to certain resources". Do you just mean "shared natural resources" as things like rivers, oceans, air, fossil fuels or does the argument really mean everything in existence because it presumably leveraged those things?
For instance, suppose I pick up a rock which everyone agrees has almost no value. I chip at it with another rock for a week until it now is in a shape of some useful tool, furniture or art. In your ideal system, who owns the value created through my labor? Do all people have equal claim to the item because it started out as a rock which was a shared resource? Do I own it? Something else?
Generally, the theory behidn the argument you refer to is that everything in existence consists of two things: 1) Shared natural resources, and 2) The application of individual labor to shared natural resources.
So, everything that is constructed as "property" has at least has a component of shared natural resources being withheld from the commons, for which some duty is owed back to the commons. Even if the shared resource has little use value in its natural state, the act of denying others the right to apply their labor to it has a cost to the commons.
> For instance, suppose I pick up a rock which everyone agrees has almost no value. I chip at it with another rock for a week until it now is in a shape of some useful tool, furniture or art. In your ideal system, who owns the value created through my labor? Do all people have equal claim to the item because it started out as a rock which was a shared resource? Do I own it? Something else?
In any system which recognizes the existence of an extraction of resources from the commons as described previously, "ownership" trends to be qualified. Most likely, you would be (presuming that your extraction of the rock was within the rules governing the use of resources from the commons) the "owner" of the tool, but you would also owe a duty in exchange for both the permanent effects on the commons of altering the rock and the temporary effects of withholding the item from the commons. There are many ways this duty might manifest, but the most common would be various taxes on value or produced income.
Only if you define an individuals freedom as "freedom to deprive others of access to shared natural resources through violence" (I use the term violence since libertarians seems overly fond of referring to government interference as "violence"; except when it comes to enforcing private property). Hence Proudhon's "property is theft".
> With Egalitarianism, you end up with policies where education is strictly controlled by one governmental body
Why? Most egalitarian ideologies want to dismantle the state and introduce some form or other of direct democracy which would preclude strict government control of education or pretty much everything else.
Marx, for example, considered the only political purpose of the state to be the oppression of one class by another, and the goal of socialism to remove the class struggle by enacting policies that would remove the capitalist class, by turning them into workers. And the natural end-point, according to Marx, would be the "withering away of the state", as in a class-less society it would have no political purpose.
Anarchists argue for a more direct approach: Destroy the state outright.
> and where Education goes to the lowest common denominator and becomes basically crap for everyone involved.
Why? The original form of socialism was the idea of a meritocracy where a central goal would be to ensure people were given the opportunity to develop personally and promoted according to skill (read up on Saint Simon).
> You end up with public health services which are subpar, with waiting lines and mediocre equipment
This is contrary to real world experiences, where the universal healthcare systems for the most part are cheaper and better than mostly privatized systems like the US.
> And this has already happened in many countries in Europe where such policies were driven to the core of the governmental actions.
So this is why most European healthcare systems consistently rank better than the US system?
> My point is that libertarian ideals are more respectful of individuals
No, it is respectful of property owners. Libertarianism is what you get when you take anarchism or communism but don't follow the thinking through to its logical extension, but single out private property for special treatment.
Somehow, according to the libertarian, we need government enforcement of one persons "right" to deprive others of access to certain resources, while we don't need governments for pretty much everything else. The hypocrisy of this doesn't seem to phase anyone.