For more background on this appearance, find a copy of Craig Tennis' "Johnny Tonight". Tennis was the talent coordinator who did the pre-interview with John and Paul earlier in the day. They knew Johnny was not working that night, and basicially wanted no part of Joe Garagiola. At one point, they even balked going on unless Tennis conducted the interview himself.
Having been alive on this planet when J&P made their appearance, I watched it...in horror. Prior to their entrance, Garagiola was saying inane things like, "Boy, am I sure going to be a popular guy in my neighborhood after tonight." Worse, Doc and the band did a medley of Beatles' songs before J&P came out...all McCartney songs.
I've often wondered if the bit about being a "cop" didn't trigger Paul's "And so I quit the police department" line in "She Came In Through The Bathroom Window".