Peterson is pretty outspoken with his self help stuff that improving your own life and setting your own life in order is not at all zero sum, it improves the lives of everyone around you. Go no farther than his last stint on Rogan and he goes deeply into this.
When he speaks of chaos, he is by and large referring to personal life issues that arise from lack of personal structure. When he speaks of heirarchies, he speaks of their inevitability, not of a specific preference for a particular form of them.
He is definitely a traditionalist, in that he finds efficacy in certain traditional modes of thinking. But the ones he finds efficacy in, economically speaking, are the ones Hayek advocates.
I think you may have missed the mark here.