So, you don't get the link to that 2004 conversation?
Here is what Hic said, though of course you don't see the context:
Chris, you're right that "smoke-filled rooms" was created about Harding, but not by Harding. From the net: At 5 a.m. on June 12,  an Associated Press reporter filed his story: "Harding of Ohio was chosen by a group of men in a smoke-filled room early today as Republican candidate for president."
The selection of Harding made the tag instantly famous. As with many catchphrases, its provenance is disputed.
Back on February 21 of that year, The New York Times had quoted a Harding supporter from Ohio as saying, "The convention will be deadlocked. .. [and] some 12 or 15 men, worn-out and bleary-eyed from lack of sleep, will sit down about two o'clock in the morning around a table in some smoke-filled room in some hotel and decide the nomination."
But that supporter, Harry Daugherty, later denied he had said it. Perhaps the phrase was planted in the interview by a reporter. However, Harding himself did bring another word into use. His slogan was to promise a "Return to Normalcy". Some at the time mocked him for misspeaking, and even today some websites say he coined the term. But in fact the word already existed, and was almost as old as "normality". There's an excellent discussion in Maven's Word of the Day of five years ago.