What were Lenroot's politics? Was he seen as progressive?
Anything of substance on Lenroot is a bit difficult to come by, but from what I've seen, he was viewed as a progressive who would have given some balance to the decidedly conservative Harding. (Makes sense, upon reflection, given that he was from Wisconsin and was a contemporary of La Follette.) Guessing he'd have thrown out the Ohio Gang unceremoniously, replacing them with no-nonsense types.
Assuming he was indeed a progressive, I don't know how well he would have gotten along with Mellon as SecTreas, and whether there would have been significant departures from the economic policies as they were IOTL. However, I'd venture a guess that he wouldn't have been too thrilled with the unbridled speculation on Wall Street, and might well have taken steps to curb it.
Prohibition represents another question: during the war years, prohibition was something of a progressive pet cause in the name of social amelioration. But by 1923, it should have been (was?) self-evident that it wasn't working very well, that enforcement was spotty at best, and that the potential for corruption (anathema to progressives) was huge. Guessing Lenroot might have declared the experiment a failure and pushed to end it as part of the 1924 campaign, thereby making his re-election a landslide.
All in all, the 1920s with Lenroot might well have been an improvement over what we know as the 1920s.