Keep in mind that the Mame film soundtrack album had Lucille Ball "singing" the title role, which is about as bad as fingernails on a chalkboard. There are other songs from the show which one hears (or at least did hear back int he days when TV variety shows were a thing). Every Christmas you're likely to hear at least one version of "We Need a Little Christmas" playing at the mall and "If He Walked Into My Life" and "Bosom Buddies" were also occasionally heard in contexts outside of the original show.
The film version of TSOM was the highest grossing film of all-time until, what, Jaws (I think) surpassed it, so naturally there would have been a strong and lasting demand for the soundtrack album.
As for Mr. Gopnick's assertion that, "By 1964, all that had altered for good; a successful original-cast album went from the place where hits always happened to a place where they rarely did”, I could mention "Send In The Clowns", "Big Spender", "Ease On Down The Road", "I Don't Know How To Love Him", "Don't Cry For Me Argentina", "Memory", "Grease Is The Word", "The Impossible Dream", "What I Did For Love", "One Night In Bangkok", "I Am What I Am", "Seasons of Love" just off the top of my head.
Yes he did win the 1962 Grammy for "Best Jazz Performance -- Large Group (Instrumental)"