Common Lisp, which the author is referring to, definitely was not organized around functions being pure. Common Lisp actually doesn't have a lot of high-order functions either. A notable exclusion was COMPOSE.
McCarthy's original Lisp was organized around purity, but that preceded Common Lisp by many decades. And, while some names carried over from the original Lisp, Common Lisp had no intention to replicate all of the intents and virtues of McCarthy's Lisp. Even before CL was made, millions of lines of application code had been written in a number of dialects, which had nothing to do with translating math or anything like that.
Common Lisp has a functional argument in every library function that takes a :test and/or :key. (Even if these are not specified, they are understood to have functional default values like #'identity and #'eql). And of course apply, mapcar and other applicator friends have a functional arguments. All these things tend to be front-and-center in everyday Lisp programming, hardly "also have" features.
McCarthy's original Lisp was organized around purity, but that preceded Common Lisp by many decades. And, while some names carried over from the original Lisp, Common Lisp had no intention to replicate all of the intents and virtues of McCarthy's Lisp. Even before CL was made, millions of lines of application code had been written in a number of dialects, which had nothing to do with translating math or anything like that.