Some shows have a show-within-the-show.
Sometimes, that meta-show is a central theme, as in 30 Rock's "TGS with Tracy Jordan". More often, it's peripheral.
Often, the meta-show is on the same "topic" as the main characters' profession or schtick -- as in House's "Prescription Passion" or Brooklyn Nine-Nine's "Serve & Protect." But again, not always -- think Community's "Inspector Spacetime" or Insecure's "Due North."
But what unites all of these shows is that, without exception, they are campier and cornier than the "original". I cannot think of one example of a show-within-a-show that takes a more serious or elevated tone than its host.
Why is that? At one level, perhaps the explanation is simply that if the writers have a genuinely good idea, they of course want to save it for the real thing. But I also think that a "better" show-within-the-show would be jarring. It wouldn't feel right; the audience (the real one, not the characters on the regular show) would reject it. A show that is less campy than its host undoes the suspension of disbelief that all shows (no matter how serious) rely upon to get audience buy-in. Once we accept that there can be, in the characters' world, more seriousness or realism, it makes visible all the ways the characters are flaunting realism. In a roundabout way, this was the dissonance I spotted in Brooklyn Nine-Nine's pentiultimate episode: once the show acknowledges that Jake Peralta playing "Die Hard" or "Speed" or whatever would actually get innocent people hurt, then the whole thing isn't funny anymore.
Anyway, no deeper insight. Just a thought I had.