The tragedy of the commons describes what happens when Western mercantilism and capitalism interact with an unregulated resource. There is a firm historical basis to it . Here's the passage that launched the Western discussion in 1833:
"If a person puts more cattle into his own field, the amount of the subsistence which they consume is all deducted from that which was at the command, of his original stock; and if, before, there was no more than a sufficiency of pasture, he reaps no benefit from the additional cattle, what is gained in one way being lost in another. But if he puts more cattle on a common, the food which they consume forms a deduction which is shared between all the cattle, as well that of others as his own, in proportion to their number, and only a small part of it is taken from his own cattle. In an inclosed pasture, there is a point of saturation, if I may so call it, (by which, I mean a barrier depending on considerations of interest,) beyond which no prudent man will add to his stock. In a common, also, there is in like manner a point of saturation. But the position of the point in the two cases is obviously different. Were a number of adjoining pastures, already fully stocked, to be at once thrown open, and converted into one vast common, the position of the point of saturation would immediately be changed".
Make no mistake - the argument here is "what shall we do with these filthy peasants." Applying the tragedy of the commons to primitive societies or sparse resources is entirely apt so long as capitalism or mercantilism will interact with it - and, as the predominant economic system in the world since the 1500s, it's worth taking into account.
Which is not to say it governs everything. Sure, Namib tribes protect their resources with aplomb at a tribal level, and look how well their approach to conservation deals with Western game hunters. Except Namibia's foreign exchange economy is 50% mining and Namibia's mining rights are fully state-controlled. This is due in no small part because their two principle exports, diamonds and uranium, have been subject to the most rapacious diplomatic and extractive techniques in the capitalist pantheon. The idea doesn't originally come from ecology, although the terminology does - arguments about the general abuse of communally-held resources dates back to Aristotle at least ("That which is common to the greatest number gets the least amount of care. Men pay most attention to what is their own: they care less for what is common)".
Graeber would point out that this is why Western capitalism is such a terribly stupid idea, and that in general, civilizations fall when they ignore the needs and desires of the normies in deference to Titans of Industry™. Ferguson would point out that for the past thousand years or so, said-same civilizations haven't so much fallen as been pushed over by Titans of Industry™ further along the Civics Tree (Sid Meyer's Civilization: mixing Catan with Excel since 1991). Wallerstein would argue that civilization has always been dictated by the Enclosers of the Commons and that since the whole point of the exercise is to steal someone else's resources, those who draw resources from the furthest peripheries are the ones in charge of civilization, QED. Kriwaczek's contribution would be to point out that the first and longest-running civilizations on the planet, the Akkadians, Sumerians and Babylonians, did exactly what Namibia is doing: socialize everything of value and let the periphery fight over the scraps.
So no, I don't think you can argue that the tragedy of the commons is a "false and dangerous myth." I think you can argue that if the stakes are low enough, humans manage collective resources just fine. Unfortunately for collectivization and resources, humans tend to organize into accumulative power structures and those structures claim the commons as their own, forever and ever amen, going back to the Code of Hammurabi or before.