...I think a "solution" has just been presented.
I just realized that some of Comedy Central's clips expire after two weeks of air -- far as I can tell so far, it's more the scripted material on The Daily Show and The Colbert Report, not the celebrity interview part of those shows.

Why?

The website sez:

What does the "expired" date mean?

Due to licensing agreements we're unable to keep this clip available on the site past the expiration date. We hope you enjoy it while we have it up.