All the more reason to have Bert and Ernie introduce the classic stuff; they're probably the oldest characters that Sesame Workshop still owns, plus they're not one-note types like Cookie Monster. I could easily see the "Things that I Remember" segments going like this:
While an instrumental version of Jeff Moss' song plays, the camera eases in to a shot of Bert and Ernie's living room. The two of them are preparing to watch home videos together; Ernie starts reminiscing about some older segment they've seen. After a brief conversation, Bert (probably irritated by some joke or unrelated question) orders: "Just put in the tape, Ernie!" The camera cuts to Ernie's hand inserting the tape; Bert clicks the remote from offscreen, and the classic material plays.