I haven't read the whole thing, but I will.
My first impressions based on what I've read so far, is the author is kinda missing some points:
The "best intellectual" and the "most influential" intellectual will rarely be the same person. Peterson is highly influential because his philosophy has direct mass appeal, not because of the precision of ideas.
This can be a good thing. The guy who has the most thorough and precisely reasoned philosophy often works in a fairly myopic arena.
Peterson provides some actionable, practical advice that counters many popular ideas as well as connecting his concepts across art, fiction, history, and everyday life experiences. This makes him interesting. A philosopher who spends 500 pages elucidating the constraints of one brand of epistemology, even when done very well, is just not going to have mass appeal. Not now, not ever.
Let's use physics as an analogy. Hardly anyone knows who Paul Dirac is. He made the most amazing math formula that has ever existed, but he's not a guy who was great at communicating with the average person. Someone like Einstein is much more approachable, his accomplishments have value that is easy to understand.
Not comparing Peterson to Einstein by any means(God no!), but Peterson is a guy who can take pretty deep topics and make them universally interesting. That is a skill that shouldn't be dismissed.
I've made similar arguments for other "Pop Intellectuals" like Sam Harris, Neil Degrasse Tyson, Sagan, and Bill Nye.
There's a certain disdain for the dilettante raconteur that I never quite get. Most people are "the average person", communication is our most powerful tool, knowledge is our greatest asset. Why critique someone for communicating knowledge to largest group possible? Especially when it knowledge not normally directed at the greater population?
Feynman understood this, he seems to get a pass, because his accomplishments exceeded his popularity.
Ok, now I'll read the article and find out if I'm totally off base.