But the real point here is that any hierarchy will inevitably fail as a mechanism for ordering society
I think that's a tough case to make.
Although we reject the identitarian "intersectional" hierarchy of the woke leftists and the pure racial hierarchy of their Third Reich antecedents, that doesn't mean society can be ordered
without any hierarchy. I would argue that hierarchy is a conservative principle, in that it directly implies that some things are better than others; some achievements command greater prestige than others, and are rewarded accordingly. The left's "equity" shibboleth opposes this reality, but in fact Brain Surgeons and Sanitation Workers are simply never going to live side-by-side, and that doesn't mean that either is "better" than the other.
The currency of hierarchy in the United States has traditionally been
currency, and the key to making the hierarchy work has been social mobility. Someone might be in the lowest quartile of earnings one year, and in the top ten percent a few years later. The identarian and racial hierarchies prevent this sort of mobility, consigning people permanently to some condition determined by accident of birth. Supposedly we threw that off in the United States when we renounced titles and ranks of nobility; the woke leftists' attempt at regressing to a hierarchy determined by birth strikes me as the most fundamentally anti-American thing I've actually seen in my own lifetime.
Jordan Peterson argues that the fundamental role of The Right is to preserve hierarchy, and that of The Left is to represent those at the bottom of it. Personally I try to reflect on the limitations of a hierarchy of economic achievement and reward while recognizing that it's probably the least harmful sort of hierarchy we can maintain, and that we will have a hierarchy whether or not we like it or even realize it.