Television

If I recall correctly, Craig Ferguson was known for going off script with his guests.

The hosts or producers typically ask the guest what story or thing they want to pitch and the host tries to bring it up.

I remember Bill Burr and Conan having a mishap on that: https://youtube.com/shorts/n7YOe8stEsw?si=bXHHKNd3pcdfggQC, it's hilarious.

I also can't find the clip but I remember Letterman interviewing John Oliver and it's when he left the Daily Show and wasn't really doing his show yet so he kept talking about his wife. Letterman and him had a bit of banter and then Letterman said, "let's go to the clip" john breaks out laughing, "I have nothing going on, a clip of what?" Letterman replies, "I don't know a clip of your wife" I love interactions like that.

I get what you're saying in terms of the "fake" interview feeling but sometimes they're nice. I like when a guest tells a story on the Graham Norton Show, the others sitting by sometimes laugh so much and seem to be having fun. An example of Jack Whitehall: https://youtu.be/UIL9CiUDHp0?si=CZGIxuf0XX_vQI-E and Jennifer Lawrence just dying laughing.

I always love the interviews when it's a host talking to someone they came up with. Seth Myers and Amy pohler or Tina fey is always great. Stephen colbert and Jon Oliver is also a great watch when they're together. They are very different from most late show interviews, which I always thought most people just saw as promos.

Sadly, the article says he's been renewed.