Not entirely a fitting analogy as power in North American societies has traditionally been based on what you might call "conspicuous donation." That is to say, where in Western society power and influence have usually been based on an ability to display one's wealth through extravagant indulgence like expensive dyes (Tyrian purple, carmine, etc.), furs (ermine), gold, jewels, castles, and so forth, in indigenous North American societies all across the upper continent power has usually been based on an ability to give away (or, in some cases, destroy) an extravagant amount of wealth through gifts and feasts (the most famous example being the PNW potlatch). So these societies essentially had a built-in system to prevent long-term accumulation of wealth since these extravagant displays were essentially designed to impoverish the person who gave them (and by so impoverishing themselves for the community they gained the respect necessary to command authority). This was probably to some extent true even of the Mississippians, the cultural complex that most imitated the Mesoamerican civs who had a view of wealth much more familiar to the Western understanding of displaying one's wealth. Based on the records of De Soto's expedition, the Mississippian paramount chiefs seem to have certainly displayed their wealth in a fashion more akin to Mesoamerican kings, but they also made extravagant gifts to De Soto that remind one more of a traditional North American wealth redistribution system.Wealth is allowed to accumulate
The kind of people who accumulate the wealth use it to buy the political process and rig the game in their favor at the expense of the kind of people who do the actual work
Eventually the accumulate parasites bleed the working class to death and everything collapses.
A second place this analogy breaks down is we certainly don't see this kind of cycle between hierarchical and non-hierarchical societies in Europe or the Middle East. Popular uprisings may bring down one hierarchical government, but they inevitably replace it with another. The Lord-Protector did not create a less hierarchical society than the Stuart monarchy, for example. At most you get little non-hierarchical communes (invariably religiously motivated) like the Mennonites or Moravian Brethren who are eyed with suspicion or outright persecuted by their neighbors.
I don't know enough about Oceanic, African, South American, or Siberian societies to say the situation in North America is unique (indeed, I know it has some parallels in Central America), but it's not at all akin to what was happening in Europe, North Africa, West Asia, or East Asia. Popular revolts there invariably led to a new hierarchical government or a brief period of anarchy followed by bloody suppression and the restoration of hierarchical government (in the latter category the peasant revolts of the early sixteenth century spring to mind).