Saturday Night Live is a sketch comedy show. Yes, there are still music performances, this seems almost done not because people actually care about the music performances (people mostly hope the musician shows up in a sketch) but out of tradition.

America's Got Talent isn't a variety show it's a reality competition show.

The issue is that it's a variety show, and people don't like variety shows anymore, because media is so splintered and the means by which people consume media is so democratized that the very idea that made variety shows possible basically doesn't exist anymore.

The Muppets trying to put on a show is a fine framework for interpersonal dramas that lead to really funny situations. The show being a variety show is a terrible choice in the 21st century. This show seems to have figured out a way around that.