As a recovering (and occasionally relapsing) admirer of TR, I’ve diagnosed my misplaced boyhood affection as the result of three tendencies that, while idiosyncratic, I suspect extend to other young conservatives:
1) As John gets at, the fact that TR was a favorite president was more about the man than the policies. Learning about the New York police days, the Rough Riders and the charge on San Juan Hill, the African safaris etc. — it was as if Ernest Hemingway were president with a big-ol’ (R.,) next to his name. But of course, like Hemingway, Roosevelt was the greatest champion of his own myth, and the reality was far more complicated.
2) In connection with (1), there is no doubt that TR still has a certain cachet among not just, say, liberal northeast Republicans, but many Democrats, independents, and even folks uninterested in politics. So, if you were like me and were educated mostly by and among liberals — and you had the adolescent desire not to be thought a complete weirdo — then Roosevelt and “Bull-Moose Republicanism” became phrases used, much like “Blue Dog” is today, to signal to the other side that you shared some of their basic concerns, even if you disagreed in the specifics. (I should hasten to add, it is far from obvious that Roosevelt Republicanism fits this agreeing on ends/disagreeing on means narrative at all. I’m just saying it worked in a pinch.)
3) As Kevin just put it to me, for a lot of us, Roosevelt Republicanism just meant national parks and trust-busting and maybe a few other eccentricities. But the continuity between Teddy Roosevelt’s governing philosophy (particularly post-1908), Progressivism, and the broader totalitarian moment has, I think, only recently entered the conservative consciousness in a big way. That’s thanks in no small part to a neat little book on the subject.