The voice I'm hearing in my head is actually from the Simpsons spoof of I'm Just a Bill . . . which I just learned is Jack Sheldon, the same guy that did the Schoolhouse Rock video 20 years before. I think his voice is better in the Simpsons one. I love that they went and got the same guy - they're so quality.
You know its an old, campy video when you hear "the Bill" threaten to expose Ted Kennedy as gay if he objects or fights back against any flag-burning amendment.