Here's the thing: I remember watching this when it was first broadcast! After my family moved to the suburbs in 1971, the television was pretty much constantly on. I'd watch Mike Douglas' show if there was a Gilligan's Island rerun I'd already seen a half-dozen times. I remember watching it and being vaguely impressed by it; he seemed really damned serious, and I'd never heard Sinatra's version.
During my search, I found more travesties. For instance, Shatner
Perhaps the worst was Shatner performing the Elton John/Bernie Taupin song "Rocket Man" at a science fiction film award ceremony, introduced by lyricist Taupin. How much do you think Taupin and Shatner needed that money? They probably paid Taupin in a couple of grams of you-know-what and Shatner got a couple of free passes to the buffet and a bottle of scotch.
The best, though, was Chris Elliot doing a straight-up rendition of that very same performance on Letterman's show. My guess is that it was in the late eighties or early nineties.