His message is that you should not eat oils because their nutritional value has been decreased in the process. He does not meditate about the different fat acids and their functions. In my opinion, he is missing out that there are nuts and oils that are quite similar in their fat composition. Of course, nuts have proteins and carbohydrates (mostly sugar) as well. But what he forgets is that you won't eat olive oil alone - you'll most likely have it with a good amount of greens, tomatoes or whatever.
Just have a look at the study he mentions when he's talking about the impairment of artery functions after eating olive oil: https://onlinelibrary.wiley.com/doi/pdf/10.1002/clc.49602214...
Quote: "This impairment, however, was also totally eliminated when vitamins C and E were given. As with antioxidant vitamin supplementation, olive oil, eaten with vinegar on a salad, did not impair endothelial function. Some societies that use the Mediterranean diet may have learned to provide the natural antioxidants which buffer the oxidative stress of these fatty meals."
He totally eliminated this aspect so he can ban the oils. Another discussion here on HN blamed him for cherrypicking studies. I'm not sure if that's wrong. He has some good advice in general but for this aspect, I don't really trust him.