'quickest': adverb

Page 442 of Collins Cobuild English Usage reads

Quick is an adjective. You do not usually use it as an adverb. Instead you use quickly.

In writing, you usually use more quickly. He began to speak more quickly.

You can use the superlative form quickest as an adverb in speech or writing .

. . . and Freedman reacted quickest to head the ball into the net.

Why is the adverb quickest allowed in writing, unlike the adverb quicker?

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.