In 1919 Teddy Roosevelt was gearing up for another run at the White House. A split 8 years earlier between he and then President Taft had divided the Republican party and given Woodrow Wilson the White House. It was almost universally accepted that TR would be the Republican nominee and that Warren Harding would be his VP. But TR died. Harding went on to be the Republican nominee, to win the Presidency, to run the most corrupt and flawed administration in American history, and to set into motion the policies which some feel led to the Great Depression.
In this ATL Roosevelt does not die in 1919. Instead let's give him 8 more years of life. So if he wins 2 terms he dies just before the end of the 2nd. Keep in mind, Harding died in 1923.
What happens in a world with TR living longer?
Let's assume all of the above happened--and by the way, I agree with the proposition that TR not going to the Amazon basin would have wound up extending his life until well into the 1920s if not beyond.
Anyhow, dealing with the easy one first: suppose TR had chosen Harding as his VP, and suppose TR died in early 1927. After the great outpouring of respect from the nation and the world (I suspect even the deposed Kaiser Wilhelm II would have found a way to attend the funeral; he held TR in extremely high regard), the presidency devolves to the Secretary of State since at the time there was no mechanism for filling a vacant vice-presidency prior to an election. Now, although they didn't care much for each other personnaly, TR and Charles Evans Hughes could work together, and they respected each other's intellects and capabilities. On that basis, Hughes becomes president in early 1927, perhaps appointing someone near the end of his career like Philander Knox to the Secretaryship [?] of State.
The 1928 election might be a bit of a challenge. I'm not sure Hughes, having tried once in 1916, would be welcome in 1928 as the nominee--although you'd have to presume he'd have learned something from his near-miss. Seems to me that TR would have had a successor waiting in the wings, and for that successor, I'd suggest his Secretary of the Treasury, Charles G. Dawes (Coolidge's VP in OTL). Dawes was a real mover/shaker in the financial world but his base was Chicago, not Wall Street, so that would make him more acceptable to the heartland. Also, he wasn't afraid to speak his mind rather forthrightly (when questioned about his bureau's accounting practices during the Great War in OTL, he responded, "Hell and Maria, we were trying to win a war, not keep a set of books!").
That said, I would suggest that Dawes would have been alarmed at the outrageous buying stocks on margin in the early 1920s, and would have recommended a series of measures to TR to bring that to heel before something really ugly happened. That would have led to the creation of the Securities and Exchange Administration, which would have oversight on stock trading, among other things. Thus, what would have happened at the end of October 1929--early in Dawes' tenure in the White House--would have been no worse than a middle-weight Panic of earlier days.
It's true that TR had little time for the Fourteen Points and less for Woodrow Wilson--and it's equally true that he propounded the concept of an international cooperative body first. Thus I suggest he'd have cajoled the Senate into buying into the treaty with modifications, and at the same time would have used his international prestige to revise the charter to include his modifications--much to the chagrin, on both counts, of Woodrow Wilson.
TR and the Soviets...now that's sticky. Still, he was pragmatic enough to realize that you can't ignore the largest (physically) nation on the globe. Could be he could have mustered enough respect to get some kind of a working relationship early on, followed by actual recognition in the 1920s. Maybe the prospect of dealing with TR rather than the unknown Harding or Cox in OTL would have been enough to prevent Trotsky from being marginalized?
While TR himself drank only occasionally (wine once in a while and a scotch on rare occasions), I doubt he'd have been a fan of Prohibition at all; probably he'd have called it a waste of federal resources, and gone about securing repeal. That wouldn't have won many friends in the Bible Belt but he wouldn't have carried those states anyhow. Repeal Prohibition by, say, 1923 or 1924 and there goes the rise of a lot of gangsters to the notoriety we know: Al Capone, for example, would have likely remained a relatively small-time hood in New York rather than the overlord of Chicago.
Further butterflies: no Hoover in 1928 or 1932; perhaps an earlier conversion of Wendell Willkie to the GOP; lower tariff walls that didn't exacerbate the economic crisis of the late 1920s/early 1930s...
