Here's something else -- say WWI breaks out in 1914, as OTL, and in 1916 TR is elected to a third term.
First, how does this affect the peace? I see an earlier deal being made not long after the February Revolution in Russia.
Main effect is that both the US and Britain are marginalised. If France still has her Russian ally, the two of them together can dictate what happens on the continent, and needn't pay much attention to what the Anglo-Saxons think. Ironically, having the assertive TR at Versailles drastically
reduces American influence there.
The big question is what happens when Russia's democratic experiment fails (a dead cert, given Russia's history) and she becomes a dictatorship of either left or right. Unlike Hitler, the new dictator won't need to spend time rearming, as Russia, a victor power, was never disarmed in the first place. Could get lively.
Second, assuming the POTUS still dies 1919, does his VP stand a good chance in 1920? (I'm seeing Hiram Johnson, FWIW...) My point is, could this significantly weaken the Denocrats permanently, forcing another realignmnet? I know RB says they'll turn to the Boubons, but I wonder if that'll save them, or even help...
If (as we seem to be assuming) TR still takes America into WW1, then if OTL is any guide his VP (or indeed any Republican) doesn't stand a chance in 1920.
After a year or two of war, Americans were fed up to their back teeth with all the rules and regulations, sacrifices, etc, and inevitably took it out on those in power. So the administration party (OTL the Dems, of course) took a hefty blow in 1918, and was all but annihilated in 1920, holding only 22 HoR seats outside the Solid South and the Border States. Had the Reps been the "in" party, there is no obvious reason for them to have fared any better.
Imnsho, we can virtually take it as read that whoever wins in 1916, regardless of party, is doomed to lose (and probably lose badly) in 1920 unless he can stay out of the war altogether - and TR is not a promising choice for that

.
If the Republicans (under TR or anyone else) take America into WW1, that for good or ill means a 1920s dominated by the Dems, much as OTL's were dominated by the Reps.