Your hugely impressive essay just made it to my Medium reading list yesterday. I am deeply sympathetic to your argument, but I do have to ask, am I mistaken in thinking that Critical Theory is a tributary of postmodernism that does in fact apply its critique to leftist as well as rightist ideology?
To my mind Critical Theory has erred in seeking after the possibility of an ideology that would not have an intrinsically totalitarian impetus. Such is not possible, I argue, because beliefs form the foundation, the “le pointe de capiton,” as you say, of every ideology, generating a Hobbesian war of all against all among ideologies in which the adherents of each must either impose it on the holders of all other ideologies of suffer the imposition of some other ideology on themselves. ‘Human (moral) equality’ and a priori ‘Rights’, such as ‘Natural Rights’ or ‘Human Rights’, are examples of ideological (secular) beliefs.
My real purpose in responding is to share something (of mine, alas) that I think you might find interesting. I have spent my adult life developing an approach to justice that is in fact non-ideological (or -theological).
It is strictly rational, containing not even one belief (i.e. an assertion of a ‘reality’ outside of — beyond—transcendent with respect to — material existence). To utilize the given place of our rational capacity in human existence in seeking a possible ethic of justice is not to ‘privilege the rational’.
If interested, I have posted a brief (therefore very dense) summary of it here in Medium: “Real Justice (summarized for a ‘5 min read’).”