Actually, to stick up for Mister King, his advice is often taken out of context. He advises using them sparingly:
"They're like dandelions. If you have one on your lawn, it looks pretty and unique. If you fail to root it out, however, you'll find five the next day... fifty the day after that.. and then, my brothers and sisters, your lawn is totally, completely, and profligately covered with dandelions." (And I think this is a sentiment we can all agree with -- adverbs, like any other word, need to be thought about, not sprinkled in, willy-nilly.)
So I don't think he's saying anything different from anyone else. He is equally down on passive sentences. The reason why he's known for being totally anti-adverb is, I think, the next bit:
"I can be a good sport about adverbs, though. Yes I can. With one exception: dialogue attribution. I insist you use the adverb in dialogue attribution only in the rarest and most special of occasions... and even then , if you can avoid it." (This one, I agree with, and I think the move from dialogue attribution to action tags is an extension of this.)
But, this is an excerpt from King, and I didn't have to look long to find it (I lifted a random King book and opened it and got this from the first page, I mostly do when this comes up):
"I wrote her a letter as soon as I hung up, hating the stilted, artificially chatty tone but not knowing how to get past it."
So, King uses adverbs, he accepts the odd one looks good, but doesn't want his grass sprinkled with them. It's very different from what is so often quoted that he doesn't want to use them at all. He detests them only in dialogue attribution. I pretty much agree (but I quite like King.)