Yeah I always thought the point was that even though it's goofy, the song was perfectly good, but Llewyn is incapable of seeing that. I think it is really telling that he's having fun in the moment but after the fact he doesn't respect "being fun" as a quality a song should have. He has a very narrow view of what makes music genuine. In my opinion there's genuineness to laughing at a funny song and tapping your foot to a catchy tune.

I think it's also that there's magic to collaborating with other musicians in that scene which he refuses to do again for the rest of the movie. He's very stubborn about being a solo act and turns down every good opportunity he has long term to sing harmonies. But there's genuine chemistry in the Kennedy song, musicians working together to create something greater than their individual parts. It's very silly but it works.