I think originally Kermit wasn't even going to be in the show as much.
I think the reason for Fozzie and Piggy not really being on the show was because Frank Oz was moving away from regular performing. Rizzo was new enough though that he should have been on the Jim Henson Hour, and I don't know for sure but I don't really remember any new characters for Richard Hunt wither. So, perhaps part of the reason for lack of older characters was simply because the performers were not there full time. You did have Gonzo there a lot as well as Kermit, and for some reason Link (I guess you need at least one pig).
It seems to me that the formula that works is established characters AND new characters. At the very least you need Fozzie, Gonzo, Kermit, and Piggy. They are the core group. On the side you can mix and match old and new, add in Pepe and Rizzo and so on. But you really need those main four in a show to keep people happy. Muppet tonight had half a good idea, but they needed Kermit to be more involved, which he was near the end, and really Eric Jacobson could have saved that show if he had taken on Fozzie and Piggy back then. The characters were absent too much.