The Doctor regenerating into a woman was set up by Chris Chibnall so it wouldn’t be a barrier to who The Doctor always was. New Doctor, similar traits, same memories, continuing adventures. The intention was and remains that The Doctor won’t be thwarted or defeated by just the fact of her latest embodiment, just as before. This does not mean The Doctor can’t mention she is a woman, nor that being a woman will be absent of consequences, especially when she drops into earlier historical periods. This is because being in a woman’s body does have consequences. It is less about what The Doctor does or says, and more about how she is seen and understood by others. As it was and remains for all people with female identities (no matter if there is a flat management style – love Graham’s line).
Thus, in King James’ time there were consequences for women. In the early modern era amid clashes between different faiths, and politics, between old ways and newer technologies and philosophies, everything was at stake. Including women who worked as healers. It was all too easy to harness community rivalries, add newly translated Bible verses with a modicum of men professionalising areas of traditional women’s work, and voila: actual stakes for women. Yes that was a cute joke, but the term witch hunt still has potency. Otherwise it wouldn’t be flung around today by people who don’t understand its significance.
So to those complaining about The Doctor mentioning gender: she does because people complained then, labelling women witches, drowning out their voices until they vanished from history, and because people still complain.