Was thinking about this the other day--it's one of the weirdest skits on the entire run of Late Night with Conan O'Brien. (Considering this show gave us Pimpbot 2000, Cactus Chef Playing We Didn't Start the Fire on the Flute, Pizza Ghost, that's saying something.)
What really makes it stand out, I think, is the melancholic ending--very unusual for a show that was build on bizarre, self-mocking humor. Conan would get teased and bullied a lot in skits, but never like this. I think that's why this particular bit has stood out in viewers' minds for so long, because even for a show that constantly pulled things out of its ass on a whim, this came out of absolutely nowhere.
(Of course, in the actual broadcast, they returned after commercial break and Conan was all like "hey they let me have the show back!" because that's just how they rolled.)
I'm not sure if any current talk/variety show would (or even could) attempt something like this--it would be too controversial and alienating.