The split in the Republican party goes back to well before the time of Teddy Roosevelt, the most prominent maverick to arise from the time of the founding of the GOP, with its “Whig” faction and the “Radical” Republicans (later the “Progressives”), divided, for the most part, on how the recent insurgency of the Confederacy should be managed to re-integrate it with the Union. The Radical Republicans won, and Reconstruction was forced upon the South, causing a very long-term situation where the South was almost solidly Democrat until the latter part of the 20th Century.
Teddy Roosevelt led the Progressives to form its own “anti-Republican” party, known in 1912 as the “Bull Moose”, and which catapulted the minority Democrats into the White House, not as Progressives, but as what in England were known as “Fabian Socialists”, the intellectual foundations of which were very much like those of the various National Socialists parties forming in countries elsewhere around the world. Egalitarianism, Fraternity and Liberty were conflated with some Utopian idea of what later proved to be an extension of the proposals of Karl Marx. The Great War (World War I) exploded all the old “normal” beliefs in the “conventional wisdom”, with Europe sunk in turmoil and despair, and the United States on a rampage of lawlessness fueled by the widespread disrespect of the law, most notably the defiance of Prohibition, the forbidding by law of the consumption of alcohol in just about any form.
Prohibition was formed in people’s minds as being due to the “old school” Republicans, but it was nothing of the kind, it was a construct of the Wilson years in the White House, which also brought us into the First World War, a fight we would have done well to have stayed away from.
The Progressives, now a very distinct splinter away from the older Republican philosophy, latched upon Teddy Roosevelt’s very distant cousin, FDR, and uncle to Eleanor Roosevelt (both her maiden and married name, didn’t have to change the monograms on the silverware and linens at all), and formed a de facto alliance with the Democrats, to pass the New Deal. The old school Republicans (the “Stalwarts”) were pretty much locked out of the legislative process for years, as Robert Taft (”Mr. Republican”) wandered largely in the wilderness, until after the end of the Second World War, a fight we could not have stayed out of, considering our support given the British for years before the outbreak of hostilities.
The drafting of General Eisenhower in 1952 put the Republicans back in the White House for the first time since Hoover departed in 1933, but Ike didn’t even know if he was a Republican or not. One does not become a general officer without playing some political strings somewhere, and Ike rose to his position under the tutelage of some pretty big names in the FDR Administration. So even if the Republicans had regained the White House, there was not much incentive or willingness to repeal a lot of the New Deal programs, and disenchantment was widespread, especially when it looked like there was going to be a state of “perpetual war” with the forces of Communism.
The Progressives, who by then had fully departed from the Republicans and integrated mostly into the Democrats, changing their party designations and everything, were also so riddled with Communist influence, they had become totally suspect in every initiative they undertook. Much of the FDR Administration had Communist agents or sympathizers at every level, and it looked like the nation was being slowly but surely subverted from within, and would fall to Communism with a slow but sure certainty.
Then Ronaldus Magnus happened. “Mr. Gorbachev, tear down this wall!” For a while, it was “morning again in America” and OK for people to say so. The Soviet Union and its empire were splintered, and the enemies of America were thoroughly discombobulated.
What has happened since? The old looming tyranny, Communism, has morphed somehow into the Islamic Caliphate, and we stand at a crossroads again.
Not really. We take the lesser evil fork every election so it’s more that we are already walking straight into our own demise.
If you're talking about big ideas -- maybe. If you're talking about the actual people who supported Roosevelt in 1912, they ended up in a variety of places politically. Some moved over to FDR and the Democrats. Others went back to the GOP. Still others gave up on the two major parties. TR himself rejoined the Republicans and may have been considering running again as a Republican.