I'm a bit confused as to who is who in the video. Is the guy with the hat and the double the PH and the guy with the bolt action the client? The video doesn't come through in great quality.

It was certainly a close shave situation. A couple of observations;

A double is good for two quick shots but if they are not effective at stopping the charge you are standing with a completely empty gun and under that sort of pressure you are not going to be able to reload the double quickly. Yes he did ultimately manage to reload his double but was very vulnerable once the bull was in close and it was only good fortune rather than good management he wasn't nailed.

The guy with the bolt action seemed to be slow or having trouble getting another round chambered and ended up stepping back into the camera man which could have been real nasty if they had both gone down. I have a copy of another buffalo charge video where the PH's bolt action would not eject empty cases, they had to be cleared by hand. I would clout a PH round the ears for that sort of thing if I were hunting with one.
A DG, or any game for that matter, bolt gun should be absolutely slick feeding and positive ejecting able to be operated without taking your eye off the animal.