Peterson is a Modernist fighting against Postmodernism. He tries to say there is an objective scientific reason society organizes itself, instead of Postmodernism's assumption that society is organized around social forces.

But the bigger issue he tries to advance is the Modern value system is true and thus good, while Postmodern value systems of social equity are bad.

I would argue that Peterson actually has a lot of 'Postmodern' ideas he is trying to suppress, like the fact that society is based on stories and narratives. Peterson needs to come out of the closet and admit he has these thoughts.

Like anyone in the closet suppressing their true feelings, he struggles against himself. His anger comes from suppressing his shadow.