people were using the off button for the B&W Minstrels, that's why it was dropped. It was never banned; variety was just on the way out. It continued to tour live, ending up doing the rounds of Butlins in 1987 when it was finally abandoned.
As for the Carry Ons, they're just cheap to show and people have been persuaded that they are fine examples of British humour. I imagine they probably are, which doesn't say much. But the infamy joke is the only one people remember, and that was nicked from a radio comedy.
British radio comedy, now that *was* good.