Do You Really, Truly, Actually Need That Adverb?

Along my quest of editing The Long and Winding Road into its eventual end product, I've butted heads with many adverbs. Some say to never use them, but they're part of our language for a reason, right? At the same time, I do see them weakening verbs and adjectives from time to time. So, if it's all right to keep them sometimes, and it's all right to nix them sometimes, when do I do what?

I decided to make a sort of flowchart for my own theory on proper adverb use, as a reference during editing.

What do you think? Decent adverb theory? What's your stance on the dreaded modifiers?