Keep in mind a lot of fans are OLD like me. They aren't sexist (though they do use sexist terms), they are conservative. They don't like change. Or more accurately, in a life of constant change, they like some certainties. And, in a way, a role model has been removed, especially in Capaldi. Despite terrible stories, he was their guy. Now he's gone, and we/they have to adapt yet again to change, big change.
How old is old? I'm 54. If you like certainties, and hate change, Doctor Who really isn't what you're looking for. If you're ignoring all the show's history before 2005, you're cheating. I guarantee Chibnall's 13th Doctor era will have a lot more in common with the 12th or 11th than either of them do with the eras of Hartnell, Pertwee, or McCoy.
Now, for guys like me, role models (and yes, we still need them) are a guide to being a better person... and we have one less. Gender shouldn't matter but it does. Who's my role model now? Frank Underwood? People become fans because they get something from the characters.
Like I said, I'm 54. I haven't looked for a role model for a long damn time. Even when I was in my teens and twenties, I read a lot of Robert E. Howard's Conan and a lot of Michael Moorcock's Elric, and neither Conan nor Elric was ever any kind of role model. Neither were the protagonists in a lot of Lovecraft's stories. Neither were the hapless losers who found themselves in a film noir or a noir novel by David Goodis or Cornell Woolrich.
And now old white guys are expected, yet again, to roll over, at the risk of being called sexist.
You mean like the way old women have put up with the white guys all their lives?
Easiest way to avoid being called sexist is not being sexist. The world moves on. You can't tell the world to put away the smartphones and bring back rotary dial wall-mounted phones, and you can't keep having men at the centre of all the TV shows.
I've always seen myself as a progressive, but I'm old now. This one is a little harder. Frankly it feels like it's something being taken from me.
It is. We're confiscating all the tapes, DVDs, blu-rays, comics, novels, toys, and games that depict male Doctors. All gone as of next year.
Actually, no. No one's taking any of that away, any more than bringing the show back in 2005 wiped out the 1963-1989 stuff. I still have some original series stories (and a lot of classic Doctor audios, books, and comics) yet to experience. Nothing is gone. Except your sense that in some way you have ownership over Doctor Who, and really, you never had that.
Just remember, some guys find it hard to change. Why shouldn't they? To just dismiss them as sexist is cruel and as sexist as you claim.
A man my age is old enough to have been a child, a teenager, an adult, a parent (well, uncle in my case), a husband, a grandparent. My best friend, who's also 54, just retired from the military. That is a lot of change.
The way we're communicating right now didn't exist for a fair chunk of our lives. I grew in places where there were a couple of TV channels, few restaurants, and a whole lot of white people; now I live in a city with tens of thousands of people who've come here from around the world, food in restaurants and grocery stores that I never knew existed until I was an adult, and an amazing array of entertainment options.
In my career as a librarian, I started working in a place that had no computers and now I rarely use books in my work. It's been 90% working with computers for years now, and that's gone from DOS 3.0 to current versions of Windows, with journeys through Novell Netware, UNIX, Mac OSes, iOS, and Android.
Over time, I've become an atheist, a vegetarian, a homeowner. My relationship with my parents has changed. I've worked in a few different places. I've worked with or hung out with people who are mentally ill, alcoholics, people from different class backgrounds, gay people, even conservatives. I've seen other people go through a lot of changes. That's what life IS.
TL;DR: it's not about sexism, it' just being conservative.
As Spock or Sherlock Holmes or somebody said, "A difference that makes no difference is no difference."