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.