Agreed with Trad about never letting go and practice, praactice, practice.
As for grabbing the rope above the device when the device is a Gri-Gri, I suppose it is possible to assert enough force to slow the rope whipping thru the device to prevent the Gri Gri from camming and clamping down on the rope. But that would also likely result in nasty rope burn (as in this case) and slowing the speed of the falling climber down significantly.
With enough rope drag, a falling skinny/light leader might not be able to cause the Gri Gri to lock right away, until he or she has gained enough momentum to "automatically cam" the Gri-Gri's lobe. Without the belayer's intervention, this would mean a longer fall or possibly hitting a ledge/protrusion. Ironically, in this situation, it might be easier to make a good catch using an ATC device (with the rope drag helping the belayer rather than working against the Gri-Gri). And yes, I have seen this actually happen at a crag. No harm done but a wake-up call for all who witnessed/experienced it.