Well, the audio version you're referring to is from an album for The Muppet Show. Kermit's introduction was recorded specifically for that. As for why Kermit didn't specifically introduce Robin singing Halfway Down the Stairs for the show, that's probably because Robin sang it in the first season, and he wasn't actually established as Kermit's nephew until the Bernadette Peters episode in the second season. So at the time, the writers and performers might not have even thought to have Robin be Kermit's nephew yet.