Having been invoked (as it were!) by Mr Fly...I can't speak for TV archives, but, as far as Radio sessions are concerned, anything from the 60s/70s has survived more by luck than judgement. The Beebs policy was that the medium (ie tape) was more valuable than what it contained. Even more so with TV tapes...Recycling was the normal procedure. In those days, very few people had video recording machines at home, so unless a diligent producer kept a copy in his attic, they all got re-used. Obviously, In this digital age, everything is scrupulously kept, from the banal to the sublime. Every now and and then, a little nugget of Film/Audio will appear, but, sadly, many fine performances have been lost forever.
I do know that there is an archiving project of what is left of Radio tapes ongoing (and has been for at least 20 years).
Whether it would ever be made available (Maida Valeable....Joke!) to the general public is highly unlikely. Occasionally, some bands will buy the rights to Peel sessions, whatever. But, it's becoming increasingly less cost effective in these straightened times. As for video tapes from that era. As I say say, They cost shed loads of money per roll of tape, and were routinely recycled after transmission.
Glad you found that John Sebastian clip..Wouldn't hold my breath for any more like it though. Sorry...