Uhura was flirting with Spock, and he was in the mood to play along (both literally and in the sense of camaraderie - or maybe it was just that Leonard Nimoy had trouble keeping a straight face during the song?). I maintain that they had a properly respectful professional relationship, and a respectful friendship (since they share an interest in music, that makes sense). At no time would I expect to see Original Spock and Original Uhura kissing anywhere, never mind in public, or while on duty.
That's your interpretation. And it's a valid one -- but so is the interpretation of Orci and Kurtzman that it's perfectly possible that the two were in a relationship.
My point was, that relationship wasn't some crazy thing they threw in there out of nowhere -- there's evidence in TOS to support the notion, primarily in "Charlie X" and "The Man Trap."