Well if Roosevelt was elected to a third term in 1908 there would be no reason to run for a fourth term in 1912. Maybe an alternate WWI could occur during his third term that would give him reason for a fourth term. But in my opinion I think if he was elected in 1912 for a third term by ousting incumbent President William Howard Taft at the Republican National Convention he would be reelected in 1916 due to WWI. Maybe term limits would be established earlier. If Roosevelt was President during WWI he would no doubt want the United States involved in the war from the get-go. So WWI might have ended by 1917 but with many more American casualties. It would be the Democrats that would dominate the White House during the roaring '20s. Maybe Herbert Hoover becomes President in 1932 or Theodore Roosevelt Jr.
In terms of reasons I would more or less use Teddys hands on approach to the country and believing there was more to be done or knowing the difference between him and taft and not finding a suitable replacement. In terms of the war he definitely would get us involved in 1915 but I also believe that he possibly could’ve won in 1916 since one thing Teddy did during his presidency would put the country through a naval buildup and strengthening the military, so more soldier would be able to get into the battleground and move in sooner than in our timeline. Taft could still influence him and continue his trust busting, along with the US pushing its influence in Latin America. The republican would continue to dominate due to the political landscape at the time and having clout with Theodore. In terms of the democrats they still won’t get the 20’s due to republican and Democrat strongholds at the time where the only two democrats who won in the string of republicans were Cleveland who was an outlier and Woodrow Wilson who won by accident. Also of course Woodrow Wilson wouldn’t have gotten elected and done things like the Federal reserve, birth of a nation, and began devaluing our currency.
Watch Alternate History Hub's YouTube video called "The Election That Ruined Everything (And If It Never Happened)". It explains how a Roosevelt presidency would've differed compared to Wilson's. One major thing mentioned is WWII never happens. So that means WWI would be referred to as the Great War and the War to End All Wars.
You say that the Republicans would still dominate the '20s but in most scenarios where Charles Evans Hughes is elected President in 1916 he gets blamed for getting the US involved in WWI. I don't see how it would be any different for Teddy Roosevelt being President from 1913 to 1921. He would still enter the war and the more American casualties under Roosevelt's administration would give even more cause for the Democrats to retake the White House in 1920. But if you are correct about the Republicans staying in power during the '20s then would FDR be able to be elected to a third term.