I think we are actually in agreement and I think this is just a quibble with the nomenclature.
As I was taught, the tragedy of the commons is just any situation where the Nash equilibrium (all actors acting independently optimally in their own interests) is not Pareto efficient (no actor can be made better off without making another actor worse off). In my mind I don't really even associate it with property rights (and I certainly don't think private property is axiomatic), so personally I was surprised at your interpretation of it being a fallacy.
As I was taught, the tragedy of the commons is just any situation where the Nash equilibrium (all actors acting independently optimally in their own interests) is not Pareto efficient (no actor can be made better off without making another actor worse off). In my mind I don't really even associate it with property rights (and I certainly don't think private property is axiomatic), so personally I was surprised at your interpretation of it being a fallacy.