I only started following DOCTOR WHO in 2000 (with novels and audioplays) and I was against gender-swapping the Doctor from 2000 to 2014. In the last three years, however, the show has introduced the concept that Time Lords could change gender and successfully sold me on it.
First, DW introduced a mysterious new character named Missy, a manipulative woman who was collecting the consciousnesses of various deceased characters. Missy is revealed to be the Doctor's old enemy, the Master, renamed Missy after the last regeneration resulted in a female body. And the show demonstrated that Missy was still the Master, the core character was the same, just expressed differently in a new form, which is what regeneration's all about anyway.
Originally, I felt that the Doctor, regardless of being an alien, was conceived as a Victorian era scientist and all the Doctors, when written well, have been written as this original character albeit in a different body. As Steven Moffat says, there's only one Doctor but with multiple faces and you feel different when you've changed your clothes or even just your shoes, so imagine how you feel in a whole new body. But that body, and that personality, was that of a man; a man who liked to show off for young girls and be the older brother and father while occasionally being flirty. The Doctor's relationships with women always made me think of him as someone who would always self-identify as a man.
But then came the Master becoming Missy and the Master as a woman was simply the Master with a different body. There was also the first Capaldi episode where the Doctor's old friend, Vastra, remarked that the Doctor, as Matt Smith, would flirt with women, but it was a facade affected to be understood and accepted as something humans could comprehend, and the idea that flirting with women is essential to the Doctor is set aside when Capaldi tells Clara that he is not her boyfriend.
And then, "Hell Bent" had the recurring Time Lord, the General, regenerate into a woman at which point she remarked that her last incarnation was the only time she'd ever been a man and she was relieved to be free of the male ego. At this point, it became clear to me that the Doctor didn't necessarily have to be a man, that Time Lord characteristics even in male-female relationships could shift without being totally revised in a regeneration; the flirting with women was only a technique seen with Doctors 9 - 11 that Capaldi's Doctor had cast off.
So, at this point, DOCTOR WHO has set up the concept of Time Lord males becoming women, shown how Time Lord characterizations can adjust to female bodies, indicated that it's in no way unusual among them, and also had Capaldi declare that he can't actually remember if the First Doctor was a woman or not and that Gallifreyan civilization is beyond concern for gender differentiation. DOCTOR WHO has done all the work to earn this change for the Doctor and done it carefully, thoughtfully and had the process take place over the course of three seasons and anyone complaining that the show is just doing this randomly out of desperation for ratings has not been watching the show very closely.