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 How does controlling who can use their own private property lead to freed markets and a freed society?

This sounds like communism not anarchy. 
 Communism aims to abolish private property. Market anarchism does support private property (I'm personally more in the Rothbard's range)

"Positions among free market anarchists can range: Mutualists believe in private property so long as it is based on personal occupancy and use; Cousins to the mutualists, the Georgists, envision a system of land-tenure in which people live in contractual communities whose public goods are financed from land rent; Rothbardians generally see rights to property as “human rights” that belong to all human beings."
https://c4ss.org/content/43554 
 Modern communism aims to abolish many other forms of "capital" such as "normalcy" (they claim normalcy is in opposition to queer).
 
 From the original post:

"The concept of private property is, and always has been, hotly debated among anarchists. Many believe resources should be available to all, while others prefer they be in the hands of the most efficient. Anarcho-capitalists assert that those who most efficiently supply other people’s needs would and should command a higher market share relative to less efficient market participants. Market anarchists agree that those who use resources most efficiently should be allowed to use their justly obtained property to accumulate more, if that’s what they so desire."

The line "Market anarchists agree that those who use resources most efficiently should be allowed to use their justly obtained property to accumulate more..." seems to suggest that Market Anarchist have some prerequisite for private property, and that there is an "efficient" class that has this privilege.