I was 15 in 1962, freshman in high school. Never watched much TV back then. I don't remember the jingle at all. Slinky..I remember it, but believe I was introduced as an older person. Still found them fascinating. To this day I'm tempted to send a Slinky down the stairs if I come across one.
Of course my kids had a slinky, but I didn't have the sense that you did of reliving a childhood experience. Slinky was just another toy, better than most of the commercially popular toys.