See, with the specific examples given...
I don't know Dr. Who. Never been my thing. Moffat takes a lot of crap, I've seen that much. But a) as I've said, people are allowed to hate things for a lot of reasons. Maybe it's true that he's just not very good, or at least not appealing to very many people (like I said, I don't watch the show). Maybe it's just a bad idea to spend years saying awful things about at least half of your potential fanbase. Who knew, huh? Either way, I'm not really feeling that he's a victim here.
Whedon, meanwhile... okay, there's a fair bit to unpack here. I'm gonna start off by saying that I am, on the whole and on balance, a fan of his work.
That said, a fair bit of his success has always related to his image. Even though he's definitely one of those writers I can fairly clearly hazard guesses at his fetishes, at the time, a show with a feminist theme and a queer woman who was only moderately fetishized for the male audience was a big stride. But you can't take a boost for your persona and then complain if your career takes a hit when the image is, let us say, tarnished.
Even then, as far as his career... he hasn't had any movies come out since the revelations, but then, the last two movies he's had his name on were fair-to-bad. He's co-creator on Agents of Shield, which is wrapping up three years after the revelations came out (a post-cancellation runtime longer than Firefly or Dollhouse lasted). There's a Buffy reboot coming out with his name on it, and he's getting another show about a bunch of women given special powers (who could have guessed?) set to release in 2021? If that's what being cancelled looks like, y'know, sign me up for some of that.
Kate Smith... okay, this is a more delicate case, and if it was parodic then that kinda sucks for herself and her legacy. That said. She hasn't been scrubbed from the cultural canon. She's being re-examined in the context we find ourselves in now; that's what culture and history are, that's what they've always been. The punishment for recording some really racist stuff a while ago is that the Flyers have decided that she no longer gets to have a monument to her cast in bronze outside their stadium,