Why does COULD WE START AGAIN exist?
It's weak and slows the momentum of the show down. Practically speaking: to give Mary Magdalene (and Peter) more of a presence in the second act in terms of vocal opportunities.
Plot-wise: Its purpose is to prevent mood whiplash. Almost all of Act Two is heavy drama, except, at least in most productions, for "King Herod's Song," which is treated as comic relief. On the original album, it worked as a break from the heavy drama simply by virtue of being a slow, simple number in a wholly different style from much of the score, but that doesn't work on stage; stage has a different dynamic from records, especially when Herod's number is one of the few that immediately calls for flashy "show biz" choreography, a real "number," as it were. On stage, it plays as a wild, raucous moment riddled with dark humor, a big socko smash, the laugh of the evening, going right into the suicidal breakdown of a man (quite literally) at the end of his rope... that's a hell of a transition, which comes sandwiched between more heavy drama.
From a structural position, "Could We Start Again..." is needed to bring the audience back into the drama, to gently remind them of what is going on in the plot, and at the same time to reveal how Mary, Peter, Simon (in the 2000 revival), Judas (in the current Broadway revival; yes, they made it work, don't ask, just go buy a ticket), feel about the turn of events. That can bring us into Judas' suicide without the sudden "Wait, what the hell was that?" feeling.
Its purpose has been somewhat obscured in current productions, because everybody seems to want to move it around, or change it, or try to give it another purpose. I've seen people attempt to use it as a post-"John 19:41" resurrection scene (usually using the additional lyrics from the Paul Nicholas/Claire Moore single version from 1992 -- by and large, I detest these mediocre, generic "pop song" lyrics, which, to me, show how little the show's authors regard the song, that they were willing to massage it into a single); I've seen at least one production that inserted the number, somehow, into the middle of the thirty-nine lashes, which is at least closer to the song's original purpose, but it pulls attention away from the main event.
I'm tired of people attempting to re-purpose it. It doesn't need another purpose; it's serving its purpose just fine the way it is.
"There is no problem so big that it cannot be run away from."
~ Charles M. Schulz
Sad. 
