Last time this topic came up, someone put me on to "The Thick of It" which is quite different but also very funny. Too bad you have to subscribe to BritBox to see all of it, AFAIK.
It perfectly satirised the government of the time. The spin doctor is one of the most powerful government figures, unelected like the civil service, and operating at the whims of the press.
Q: What's the Danish term for "spin doctor"?
A: Spin doctor
One thing that kept them fresh was inviting guests-of-the-hour; often beleaguered under-fire politicians, who they would then call corrupt or liars or incompetent to their faces. Not something you'd ever see on US television, other than Stewart or Colbert.
Background: https://en.wikipedia.org/wiki/Have_I_Got_News_for_You
Episodes: https://www.youtube.com/results?search_query=have+i+got+news...