...but I understand that in that final episode there's a "secret" Doctor played by John Hurt? There's speculation that's the 12th Doctor.

Granted, I have no idea who they're talking about, as I have not seen the episodes, but fine.

Counting against him is both that he is not relatively unknown, which seems to be a pre-requisite for starting out as a doctor, as you pointed out, plus the fact that the Doctors seem to get younger each time they regenerate. (Although age seems to be irrelevant to Timelords, so there's that.)

It would be fun to have a woman, but don't Timelords have genders, too?

(And no, the ginger joke is more fun to keep going on.)