I'd forgotten that there had been some off-the-cuff discussion when TR tried to get a volunteer forces bill through Congress in 1917 (I think): namely, that Harding would introduce and back such a bill; allegedly, in the heat of the moment, TR spoke enthusiastically about Harding as a future running mate at the time.
Now, whether or not that would have been borne out in a more sober moment in 1920 is up for grabs, but I have to wonder if TR would have stood still for a running mate around whom as many rumors (some black ancestry-proved unfounded-and a mistress/illegitimate daughter-generally credited as true although long afterward) swirled as Harding. Certainly TR was a reasonable if not quite astute judge of character, plus he had the perspective that at any moment the vice president could be called upon to step in. As such, I submit he'd want a running mate in whom he'd have some confidence that the job would be carried out competently--and I doubt upon sober reflection he'd have that confidence in Harding. Someone like Lowden or Lenroot, no question, but not Harding.