My feeling as regards a female Doctor is...well, the internal consistency of the show. We know there are Time Lords and Time Ladies, and we've never been shown or told, in all the incarnations and episodes, that it is possible for a Gallifreyan to change sex when they regenerate. The whole thing with Missy, we've never had that explained, or how she can even be the Master when we saw the Master fail to regenerate and the Doctor burn his body. So there's some things that need exploring there before it can be used as evidence in the argument.

I think there needs to be a clear divide between wanting a female DOCTOR and wanting a female Time Lord/Lady. My feeling is a bit like Bond. Let's have a gender-flipped Bond! Why not do it? But don't make it the SAME character in the same universe with the same canon and expect us to buy into the internal logic (or lack thereof). Don't give us all the same backstory and history and motivation and personality and expect me to believe he's suddenly spontaneously changed sex. Give me a spin-off, give me an AU, give me a gender-flipped version. But don't just twist the whole decades-long canon and expect me to believe that Bond, or the Doctor, can just suddenly spontaneously change sex, because that's a suspension of disbelief too far.

Besides, what is River if not effectively a female Doctor? If they did try it, the new Doctor would only suffer in comparison...