First and foremost: The Doctor is a fictional character. Doctor Who is a form of entertainment. I watch it to be entertained, and I enjoy it because of the stories, the actions of the characters, and I believe in the type of hero The Doctor IS. The promise of the Doctor? "Never cowardly or cruel. Never give up. Never give in." The Doctor is a hero that saves people, saves the world, and sometimes saves the universe. Multiple universes even.
All of time and space - everything that ever was, or ever will be. The Doctor chooses companions to show this to, and helps others along the way. None of this is impacted one way or another by The Doctor's age, race, sexuality or gender. In fact, we've been shown multiple times that the nature of the Time Lord is fluid. Other Time Lords have been shown to swap gender or race upon regeneration, so why is it such an upheaval if The Doctor does it too? Seems like the sort of thing he'd do just to shake things up a bit, or pay homage to the now deceased (apparently) best friend, Missy/The Mistress/ The Master. The doctor's face has changed from old to young and back again several times now. Why is this such a big leap for people? It's always the same person. Who you are at the core, this isn't going to change.
I've always cared more about the writing. Make me feel, make me *believe* the story you're telling. And frankly, as a woman in my 40's, I relate more to the Doctor than the Companions. River Song and Missy were great, but they're pretty much *gone* now. When you're daydreaming about time travel, it helps to have an image to reference, someone you can relate to. Preferably not someone who is likely to be killed off, ditched, or otherwise left behind. It's nice to see that we, too, have a chance at being The Doctor.