It is fantastic how the kids are responsible, but grandpa and the bear trader are just along for the ride. Where is their pride as an adult? But on the other hand, the kids obviously do just fine by themselves.
As long as you're not throwing finished cuts on the ground it should be fine. Not that the author likely got it 100% right, but a little dirt isn't much of an issue.
Skin and gut the animals then throw them in the river to get the worst of the debris off. Once fully cooled, and they are ready to head back to town, the carcasses would be washed with clean water and possibly alcohol, hung to age for a few days. Only then would it be parted out into cuts.