I'll disagree. In 1900, even as a vice-presidential candidate, TR overshadowed McKinley to no small extent (there's another thread where I posted a cartoon from 1900 to that effect). Moreover, within months, if not weeks, after assuming the vice presidency, a "TR in 1904" boom started. Had McKinley not been shot, I contend that, given what already existed in support for TR, plus the tenor of the times, he would have been all but a lock for the GOP nomination in 1904--with a conservative running mate like Fairbanks, to be sure--and an odds-on favorite to win the election.
What this really butterflies away is a Taft presidency: I could somehow see Taft refusing a Supreme Court appointment in 1902 when a seat became vacant, but not when another came open in 1903 (the seat to which Holmes was appointed, IIRC). Thus, Taft comes home from the Phillippines, takes his seat on the Supreme Court, and is not within bounds for consideration for the presidency in 1912.
Who might have been the GOP nominee in 1912? Tough call. Not sure that Hughes would have been in a position to be nominated, and Root would likely have pleaded his health (he did so in '08 in OTL). Lodge's appeal was purely regional. I'm guessing MO governor Herbert Hadley, with possibly IN senator Albert Beveridge as a running mate, in a thoroughly progressive ticket.