The Muppets wasn't a kids' show, it was aimed at a broad family audience (does such a thing still exist?) and screened in prime time. Sesame Street was for children, as was the spinoff Muppet Babies. That's not to say the Muppets would go down as well now, but I can't see why they shouldn't as the format, about putting on a variety show, is pretty adaptable.
As long as they don't swear.
I've always thought Miss Piggy's 'Pretentious? Moi?' is the funniest two-word joke in the language.
(The funniest one-word joke is 'lysdexia'.)
As for Spitting Image, I don't know - that depends a lot on the quality of the writing rather than the quality of the puppets. Have I Got News for You, Rory Bremner etc cover a lot of the same ground.