This is actually not true at all. Where you claim he is challenging an orthodoxy of the left, I'd argue he is instead constructing a shadow concept, a windmill of sorts that he can tilt at with great furor. He less challenges leftist orthodoxy as argue that there is some fabricated marxist movement that has power. However, the truth is he's not presenting new ideas. He's repackagaing the status quo in a sort of verbose package that allows the norm to sound not only new but not present. The greatest trick is arguing for basically what already is, while claiming what already is is not, and he creates a sort of fear of a fanatical is, a reality you can say, that in fact is not. This is that is not he then uses as a sort of aggressive strawman to essentially repackage the real is as a sort of new mode of being. Essentially, he convinces his followers that reality is not, and through his verbosity encourages them to fight for what already is under the false pretense of it not actually being.
That is to say, he takes the status quo, the current being of the country, and obscures it, transforms it, and hides it behind what I'd call a false shadow. This shadow, which he'd deem to be the threat of radical postmodern neo marxism, that I remind you does not in fact exist, he claims is looming over us, blocking us out from enlightenment. The trick then is that he presents our actual reality, our actual status quo, as an antidote to the false shadow which he claims real. He claims to be the light, the truth, the illumination that will chase the shadows away. This of course could not be further from the truth, he is not the light, far from it. What he might call a new enlightenment is in fact as I've stated the status quo. Far from lighting the path and beating back the shadow of postmodern neo-marxism, a neologism that frankly if one with even a passing knowledge of those independent concepts would realize is a form of literal absurdity, what Peterson does, not unlike a certain Charles Montgomery Burns once did, is seek to, through his teachings, his writings, his auditory presentations, his videograpahic treatises, and his socio-digital communications, block out the light of progress. For you see the shadow hanging over our society is not post-modern neomarxism, the shadow is the status quo. This shadow obfuscates inequality on every intersectional axis, when we are blinded by the dark we fail to see the truth of injustice of discrimination. So, what Peterson attempts to do with his argumentation is essentially block out the light of progress, make it more difficult for those who listen to him to recognize injustices, he seeks to make the status quo not the shadow, but the whole. He seeks to render no differentiation between the shadow and the light, he seeks uniformity in conformity, where if injustice is ever identified no distinction can ever be made between an empowered class and an oppressed class, where for example the claim of a war on Christmas is seen as equally credible as a Muslim immigrant ban.
It is funny he rants about about radical left, but what he is is the radical norm, the radical status quo if you'd like, and that radical status quo is far more powerful and dangerous than the postmodern neo-marxist windmills he tilts at.