Think it only happens to other people? Here at The Castle, our Duty Managers floorwalk to monitor climbers. We record every incident in which we have to intervene. Sometimes it's a very minor thing like tying into your belay loop instead of the tie-in points on your harness. Sometimes, we see much more serious incidents that, if it were not for our intervention, could have led to serious injury. We publish the results of our monitoring on the Safety Board across from Reception so that climbers can see what common mistakes are and learn from them.
There are three things from the Petzl video that I think are particularly important to note. The first is that it highlights how 'lazy practice' - i.e. not securing the brake rope all the time - can quickly and easily lead to a ground fall. I've seen many belayers simply shuffle their hand up the brake rope instead of properly securing it. However, when a fall happens, it happens fast and if the rope starts to go quickly you could find yourself with some severe rope burns before you stop the climber (if you can).
Second, many people mistakenly think that autolocking devices such as the GriGri are foolproof. They're not. Even when they're locked off, the belayer should keep a hand on the brake rope in case the climber releases the camming action by pulling back onto the wall, as Nina demonstrates in the video. Again, don't underestimate how quickly a fall happens. If you're holding the cam open, you might not be able to lock it again before your climber hits the deck. The best solution? Just belay in the approved manner as demonstrated in their best practice video and the poster we have on the stairway.
Finally, while this is less applicable to indoor climbing, tying a knot in the end of your rope should be an automatic reflex. My partner and I make sure that we always have the end of the rope tied off to our rope bag whenever we're cragging. If we're on a multi-pitch then we're both tied in all the time. Most professional climbers will know someone who's either abseiled or been lowered off the end of the rope. The results can be fatal and it is a completely avoidable accident.
If you're concerned about someone belaying badly, please inform the Duty Manager (the ones walking around with the radios) immediately.