I remember the SNL appearance, and before that seeing him do it in concert. Teac tape deck on stage with its own spotlight. It started and stopped on the choruses (i.e. wasn’t running through the whole song), presumably manually activated by a crew member with a wired remote (there were no infrared remotes back then, children).
I never heard any authoritative reason why he did it. My best guess then and now: because they could. Showing off. I was pretty impressed with the precision.