OK, I will try my best... still haven't watched the video. Here in Finland, traditional method is to gralloch at the shot site. I think this is because most of the hunting has been either small game or moose. With small game it makes sense to do the deed before putting it in you backpack, and with moose it's a sensible choice even when you have dozen or more men joining the hunt. Also the concept of first class cooking and involved hygiene has been missing to great extent (meat is meat and it's put to pot and heated until ruined).
I don't have extensive experience from several countries, but I've understood that moose is the only game that's gralloched on-site in countries that have long traditions of effective drive hunting (and more game than Finland). Maybe big reds also. Reason for this is, the hunt is concentrated effort, you don't want to waste time (there's several dozen attendants) and there's manpower etc. to transfer the whole carcasses to larder. There also might be some division of duties, like pegs, beaters, transport and larder. The beasts that are gralloched will be either kept as intact as possible, (zip) tying esophagus and rectum. Or split the neck (not sternum) to get whole esophagus and windpipe out, and also tunneling the rectum. Feet usually cut, or sled used. Carcasses to be gralloched might be first transported closer to road, e.g. if sufficient "motor power" is available or extraction involves crossing or using a river.
In Southern Finland, where whitetail deer, roe and to some extent wild boar have sustainable stock, the approach is mixed. In areas where whitetail hunting has been established over several decades, usual approach is to (suspended) gralloch at larder. It's least work intensive and produces cleanest carcasses. Basically all meat is going to personal consumption, so the beast is first skinned and then gralloched. Some hunters, especially from farther north, prefer to gralloch at site. I don't find this very constructive, in addition to above reasons you probly have temperature below zero (Celsius), and it's just messy with very limited upsides.
I think most UK stalkers have optimized their process for food chain i.e. not skinning the beast. Btw I found interesting concept from Donington Deer Supplies where you mechanically skin from front of the carcass towards back. Experienced hunters do it like that anyway, but by hand and it's kind of awkward. Starting from back towards the front will tear meat and fat along with the skin. Boar are a different story, you have to fight for each inch of the skin with your knife...