A sad day indeed.
In the article it says, the partner "disconnected himself from the rope so William could rig the rope for the rappel". That would explain how the rope could have fallen with the climber and not dragged the belayer over too.
I agree that we all need to remember that viligence is vital. Moreover, I think it is important to remember that most climbing is a team effort - between belayer and leader - if the leader gets into trouble he/she may not have the mental focus to avoid making mistakes. That appears to be the case here, though the actual steps that led to the accident are unclear.
I'd suggest that when possible, the belayer should lower the shaken/injured leader to the belay to assess the situation and recover. At that point the team might decide to go back up and try again, send the other person up to try, or pull the rope and leave gear and rappel from the anchor. Either way, it allows the team to make a joint decision rather than put all of the responsibility in the leader's hands.
Generally, it might be wise at times of stress to put more trust in our team members, even when they are neophytes, because at least they have not experienced the emotional and physical trauma of the leader fall, and thus may be more level-headed.
Hopefully that makes sense.
My condolences to those involved.