

How did you search archive.org? I see several hits on https://archive.org/search?query=subject%3A"Craig+Kilborn"
For example this full episode: https://archive.org/details/kilborn.-0743
A fair number of them are “streaming only” due to them being some sort of agreement between archive.org and CBS.
I also see some on Youtube, https://www.youtube.com/watch?v=YR1fJIqpVrI
BitTorrent does seem to be completely dead. bt4gprx is a very thorough DHT search, so it will find basically anything on a public server that’s actually being seeded, https://bt4gprx.com/search?q=Craig+Kilborn and it has no hits for the Late Late show, (but a few other appearances by Craig Kilborn on other shows.)
Presumably CBS/Paramount does have the recordings in a vault somewhere… but it’s probably not worth it for them to put them on Paramount+ (Who knows what will happen when they go bankrupt). As it is you’re limited to the few episodes individuals bothered to tape, and then digitize.