Economies are made of economic actors, which includes ALL people as well as other entities like public institutions, companies, charities...
> some of whom do not make resource allocation decisions (because they are unable to or prohibited)
Every single person in the world is an economic actor. Every single person in the world has the ability to make resource allocation decisions, even if they are only using the labor of their own mind. The incarcerated are economic actors, even literal slaves would still be economic actors.
> most forms of socialism (theoretically, anyway) are democratic, in which allocation decision making is given to the people
A theoretical “democratic socialist” country (theoretical because people have voted socialism in, but never in history have they been allowed the privilege of voting it out) would be a country where the people surrender their natural rights to make their own resource allocation decisions (or at least a significant portion of those rights) to the government.
> The "ultimate free market" very clearly has a "system"
It doesn’t at all. It’s just the way humans (and really all animals) behave in absence of governance that prevents them from doing so. Ancient people most certainly had private property, even if it was shared within the tribe or family or whatever other format of social organisation. It would also pay for you to remember that Marx thought familiar or tribal affiliation was a source of evil, as families tend to care for each other, which he considered to be terribly unfair to anybody who didn’t have a family to care for them.
I would very highly recommend that you at least learn some basic economic concepts (what an economy actually is would be a good starting point) before you go around promoting the abolition of private property.