The prospect of TR being healthy enough to run for President and secure the Republican nomination in 1920 has been discussed before a few times.
A few necessary preconditions discussed have been his son Quentin not being killed in WWI and/or him not doing his "River of Doubt" exploration of the Amazon.
Let's say that happens here.
Then, we have one of the iron laws of 20th century US politics, the 1920 election is an inevitable reckoning and rejection of whatever administration led the US through WWI. Wilson and his works, including Democratic nominee James Cox, will be repudiated.
Combining the two, it looks like TR could become the challenger to the incumbent Democratic Party and would be favored to win, simply because of backlash.
However, TR and Warren Harding could not be more different. What would TR's rationale for his own Presidency be and how would he contrast himself from the Wilsonian Democrats?
"Normalcy" does not seem to fit TR. Withdrawal and isolation from the world, getting out of the way of private business, and restoring civil liberties do not seem anything like TR's top three passions.
What is TR's campaign about?
What is his Presidency about, assuming he lives a full term, or is at full capacity for at least half his term?