IMO, the era you speak of can be divided into three parts: (1) 1896-1910 with Republican domination, (2) 1910-1918 with the parties fairly evenly balanced, and (3) 1918-1930, with the Republicans again having the advantage.
It is of course conventional to say that the only reason the GOP didn't do better in (2) was the Taft-TR split, but don't forget that the Democrats won control in the House in 1910, *before* the GOP-Progressive split, and won in 1916, despite the Progressive Party being dead, and Hughes having the support of both Taft and TR. And the argument that Wilson only won in 1912 because of the TR-Taft split is based on the assumption that the Taft and TR votes could have been combined for one Republican candidate, which I find dubious for reasons I go into at https://groups.google.com/d/msg/soc.history.what-if/NZXQky_Jgpk/OhWnTSyjfwoJ and
https://www.alternatehistory.com/fo...lose-the-1912-elections.342514/#post-10256703
That being said, TR if elected in 1908 would no doubt have been a more politically savvy president than Taft, so the Republican candidate in 1912 (almost certainly *not* TR) would have a good chance of winning. But it would not be guaranteed; after 16 years, "time for a change" sentiment could be a serious advantage for the Democrats, who will have a number of candidates likely to do better than the controversial Bryan or the conservative and colorless Parker. This would be particularly true if the GOP nominates a conservative like Root, who will be easy to attack as a "Wall Street lawyer."