I think most people think of characters like Elmo and Cookie Monster singing and dancing around. I think for the most part the mainstream audience has a very limited view of what the Muppets are.
I agree that the mainstream audience's view is limited, but I don't think it is as severe as you propose.
I think they could identify Kermit, Miss Piggy, Cookie Monster and Big Bird easily as Muppets.
Mainstream audiences with a bit more attention could probably ID Elmo (I think his star has faded a bit - but I don't follow SS, so maybe he is in the top tier; Oscar, Beaker and the Chef; The two old guys - maybe they know their names, but they know their faces; and oddly enough Mahna-Mahna and the Snowths - again not their names, but I think if shown a clip they would be ID'd as Muppets.
I don't think the mainstream audiences would be aware of the different corporate sides Disney, Sesame Workshop, Henson.
Aside from the characters I imagine they would associate them with kids programming or perhaps the phrase - they did some weird things could pop up.
I have the feeling that they may not think about the Muppets or even miss their presence, but when they do see them in an ad or a cameo or a new viral video they are happy to see them.