In short, adverbs all too often do the opposite of their modifying purposes. They can weaken rather than strengthen other parts of speech, particularly verbs.
Brilliantly worded might be a tell in many contexts, but without the adverb there's no expressed comment on worded. One of modifiers' strengths is expressing comments. Marsha's brilliantly worded darlings fell flat upon the audience's ears. Might the brilliantly adverb used in the adjective phrase be an overstatement in the style of verbal irony?
Quickly ran could be construed as causing a tautology between the two words. It also might be a weak predicate phrase and weak word choices when there's so many one-word verbs that mean much more. Jogged, cantered, galloped, dashed, etc., and phrases, like ran headlong.
Conversely, adverbs play an important role in "expressing some relation of manner or quality, place, time, degree, number, cause, opposition, affirmation, or denial, and in English also serving to connect and to express comment on clause content".* Adverbs also play an imporant role in hedging or hyperbolous speech, when used in dialogue to characterize characters or for overt narrators expressing comments, for example.
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