One possibility: in the reality before McCoy arrived, the bum who got fritzed by the phaser was originally the catalyst. He was in that sport, possibly drunk and stumbling in the street in the path of the truck. Nobody else made a move to save him except Edith. She failed, sadly and they were both killed by the truck with bad breaks. This would then remove the bum from the timeline.
That's pretty good. I never thought of it that way.
This story is a test of the main characters and particularly Kirk: will he be selfish or act for the greater good?