That's a solved problem in professional sports broadcasting. They have dozens of cameras, and if something interesting happens away from the current camera they just do an instant replay from the camera that caught it best.
It's not quite the same in team FPS games, as most sports (that I can think of) tend to have one area of action at any one time, for example in soccer there may be 22 players on the pitch but there's only one ball - so replays to show what they missed are more to show better angles or other tactical plays, rather than missed action. But yes, this is still a useful (and currently often underused) tool in esports broadcasts too.
