'quickest': adverb

The rules here are not hard and fast. You'll note that the usage guide said that "quick" is "not usually" used as an adverb - which is different from saying "never". You'll also note that the usage guide drew a distinction between spoken usage (in which it is acceptable to use "quicker" to mean "more quickly") and written usage (where the authors judged that "more quickly" should be used instead).

"Quick" can be either an adjective or an adverb, but its use as an adverb is marked "informal" in some dictionaries.

The interesting thing is that, as you point out, the authors of the usage guide judge the adverbial use of "quickest" (to mean "most quickly") to be more acceptable than the adverbial use of "quicker" - finding "quickest" acceptable in writing as an adverb, but not "quicker".

My intuition is that the authors are correct that "quickest" (adv.) is less informal - and so more widely acceptable in written contexts - than "quicker" (adv.). But this is more of a stylistic preference than a syntactic rule.