Coolidge was not an inspired choice.

One thing about Harding.  Everyone remembers the two crooks, Fall and Daugherty, he had in his cabinet and rightly.  He also had a future chief justice, Hughes, and a future president, Hoover.  Coolidge replaced the disgraced Daugherty with Harlon Fiske Stone, another future chief justice.  A poll of judicial historians in 1971 rated both Hughes and Stone as among the nine greatest justices.  On balance, quite a cabinet, with  some of the best and also some of the worst.

The Pud and the Dud.  Given that Harding was one of the worst, if not the very worst president in history, Coolidge looks positively brilliant by comparison.  When Harding conveniently croaked before having to face the likelihood of impeachment, and made Coolidge president, it looked like a great choice.  This despite the fact that Coolidge was probably the most boring man to occupy the Oval Office.