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Because, okay, we're dealing with three really basic (and complementary) character types here, the Master and the Doctor and the Captain (because the Captain's always there somehow, the Other to their Rassilon and Omega, and that the Master hadn't planned for). The Master is all about *mastery*, about control, about - well - craft. Even at the Academy, he had to be the best at what he did.
The Captain? Jack's in it for the friendship. Well, and the flirting. But, mostly, he just cares about people.
The Doctor likes to tinker. Oh, he likes to think that he improves things by tinkering, but it's the tinkering, the messing about, the opening it up to see what makes it go and then putting it back together, only a bit different - that's what the Doctor does.
Fanwriters can be divided into Master types, the ones who concentrate on making it *work*, making it better, craft and excellence and elaborately worked out schemas and hard work and natural genius; into Captain types, the ones who are in it mostly for the socializing and the squee and the boykissing; and into Doctor types, the ones who can't resist the temptation to tinkering, to take canon apart and put it back together, and who cares if it looks like a piece of junk and you had to reverse the polarity of the neutron flow, watch at all the cool stuff it does now!
I'm mostly the third type. Obviously.
(Also obviously, nobody writing on the internet is only one thing - you have to care at least a little about the mastery, or you wouldn't bother with a story; you have to care at least a little about the socializing, or you wouldn't bother posting it; and you have to care at least a little about the tinkering, or you wouldn't bother with canon. Even the characters aren't all one thing. But I'm still mostly Doctorish.
Also, all the Dr. Who characters are pretty much completely mad, but you have to be a bit mad to be in fandom anyway, so that works.)