The bright line was drawn his last night on NBC, back in 1993, when the wry significance of this song was lost on no one.
Even Paul Shafer and The World's Most Dangerous Band (as they were then known) seemed a little stunned to realize that they were doing a valedictory performance with Springsteen. (Watch the guitarist and bassist gradually figure out that they're getting camera time with The Boss.)
Letterman has an insufficiently appreciated record of bringing out some amazing musical performances on his show. For some reason--perhaps not so surprising when you think about it--his forte seems to be in capturing performances with a prickly undertext, like this, this, and this.