The one concrete criticism I could see in the article (aside from vague handwaving about "pedagogical underpinnings") was about providing a step-by-step series of instructions for solving a problem without real understanding.
But Khan doesn't actually do that. In my experience, he's very good at emphasizing why something works. He encourages people to remember the "why" well enough to work out the actual formulas for themselves, rather than just memorizing the formulas.
Personally I don't understand how one would go about teaching math without using step-by-step instructions, especially considering his example of teaching slope. He never says what would be a better way to teach slope than the textbook/Khan methods, he just claims (without submitting any proof) that those methods are completely ineffective. Well, it seems to be working for KA students, and it seems to be working for other countries.
I agree with pinchyfingers: Students are getting worse at math because they don't want to put the work in. And I don't think it should necessarily be the educator's burden to make each individual math topic more fun and exciting to try and trick students into using their brains. We need to find a way to get kids motivated about learning in general again.
But Khan doesn't actually do that. In my experience, he's very good at emphasizing why something works. He encourages people to remember the "why" well enough to work out the actual formulas for themselves, rather than just memorizing the formulas.