I know nothing of how Khan Academy works and have no specific opinion on whether the author of this blog was right or wrong to call them out... But, as a general rule, I find it hard to fault people for turning to public shaming quickly as a way to get through to a tech/web company about some problem they have these days. It is, sadly, often the best path to results.
There are amazingly large swaths of the software/web/services industry where customer service and back channel communication to decision makers in the company essentially doesn't exist. Phone numbers don't exist, appropriate email addresses are hard to find and are likely to just robo-response you (and point you to a FAQ that doesn't cover your situation) if you do manage to find them, etc. Maybe there is some obscure mailing list the user could be heard on, except that mailing list is only known by experts on the software or service and good luck finding it if you aren't even aware it might exist. Or maybe there is some specific person they could tweet at, but not being part of that culture they have no idea who that would be.
I mostly blame Google for the sad state of this. Though they are hardly the only offenders, they were pioneers in this area and are certainly one of the largest who still practice it across most of their products and their continued practice of it legitimizes it in the minds of other web companies.
Again, I'm not saying any of this does or doesn't apply to Khan Academy. I'd guess based on the community nature of the product it is not nearly as bad as I've seen elsewhere, but I think the sorry state of customer service across the industry is directly responsible for these sorts of public shaming posts and the people making them aren't being irrational all things considered, the service/software companies as a group are bringing it on themselves.