The basic problem with a female Doctor
Suspension of disbelief.
You wanna write a female Doctor, first and foremost you need to write a female character.
The moment you already have an established character, that's far from easy. Even though every Doctor is different, all of them share some common psychological traits... and many of them are typically male traits. The 'average' Doctor personality is the classical 'Eccentric and courageous British male scientist' archetype which was relatively common in 50s sci-fi. Quatermass, for example, or Van Helsing in the Hammer movies.
So how do you get a female Doctor?.
Option 1 (easy one). You can write a male with boobs. This is actually the default female lead in the SJW/NPC genre.
Option 2 (difficult one). You can change essential traits to get something that is The Doctor but is not The Doctor at the same time. It would feel like a new character, related, but new, a kind of middle point between the original series and an spin-off. And that could be interesting... but it requires skill and willingness to take risks, both things lacking in modern post-meritocratic BBC (and mainstream TV in general).
You can have diverse characters and skilled writers... or diverse writers. You can't have it all.