Weirdly, I don't think you would have been called a conspiracy nut.
It was a fairly common view that Jimmy Savile was a dodgy bloke round my area in the 70s and 80s. He was often in our area and young people, particularly girls, were told to keep away from him in that specific Northern warning style where you say something fairly innocuous but in a loaded tone with a certain look in your eye.
I definitely remember that people couldn't understand why the BBC still had him on JFI in the 80s because, by that point, it had become blatantly obvious there was something not right. But, by that point, JS was appearing on national TV wearing a string vest under his open shell suit jacket, smoking a cigar with kids on his knee.
Sometimes, I wonder whether the BBC thought that was what Northern men were like, and JS was seen through that lense.
It is worth remembering that the notion of "duty of care" is a fairly recent development though. It wasn't quite the same in the 70s and 80s; these were the days before crb checks where, on a local level, such protection policing was a lot more "informal". Of course, those "informal" mechanisms that stopped "that odd bloke" getting involved with scouts on a neighbourhood level didn't scale when it came to national institutional entities, such as the BBC or the church.
I'm afraid that I view the BBC Jimmy Savile scandal as another example where the majority of the Public is very aware of a problem, but a public institution can't see it for toffee or simply ignore it because it is so inconvenient.