The Savile era of TOTP was before my time, so I can't say how easy it would be to edit him out of it. That one does make more sense to me though, as it was his show and it was wasn't something that was originally designed with repeat viewing in mind.
I'd compare it to Animal Hospital, which I did love as a child, but which you can't really separate Rolf Harris from, and was very much designed to make him seem like a lovable figure children would feel safe around. He was this warm-hearted Aussie grandfather who drew funny pictures and always had a cute new pet to show you. It was a whitewash of him as a person, just as TOTP was for Savile. I wouldn't feel comfortable showing it to kids today, knowing that.
I do think the Doctor Who question is different though. That's a work of fiction designed for longevity. DW and HE have very minor roles, and are portrayed in a neutral or negative light. I don't know if it's right that everybody else's hard work on the show, and the value the themes still have to offer young viewers, should be thrown out just because these of guys. It's losing something 99% good for the sake of 1% bad.
The show also has a significant adult audience, who are able to make that judgement call in their heads. I do the same thing when watching the first series of The Thick Of It, for example. Or when I'm watching the Lord of the Rings movies and see Harvey Weinstein credited among the producers. Engaging with this content doesn't make me think any better of those men. It's just one of those things you have to weigh up and deal with. I would feel differently if the movie was a biopic of Harvey Weinstein designed to whitewash his image, though, or if The Thick Of It hadn't changed up the cast after that first season.
For me, there are different categories to all this, and it is something that should be approached with nuance. The Huw Edwards Doctor Who episode, for example - that was pulled off iPlayer for a long time while the BBC tried to edit him out of it. The episode gets derided as a bit of a dud among Doctor Who fans, but it's worth pointing out that at its core, it's the story of a young black girl and her mother trying to recover from domestic abuse. That has value. There will probably be families, and children, who that episode helped. Was it right to remove it entirely, because a news presenter who briefly appears in the background was involved in a scandal? Does the good outweigh the bad in that? I'm not sure.
There's also the question of the entertainment industry being so awash with terrible people that once you start censoring content like this, it just goes on and on.