McKinley is not assassinated in 1901, and serves out his term. Roosevelt, while popular, would have faced an uphill battle for the nomination. So, assuming he doesn’t get the nomination (if only to change things up a bit), who does each party nominate in 1904? And then, for the re-match in 1908, once there is a new incumbent?
This decade and the next are a bit of a crossroads as each party copes with and adapts to the force of the Progressive Movement, and any variety of different trajectories might be taken.