When you try to use your commands on them at these times they don't listen, and they're not quite the same dog that they are when you have them in a quiet room or outside by themselves. Suddenly this dog that you were working with and are proud of its accomplishments has turned into a wild, running, biting uncontrollable animal. They are getting their energy out and exercising and having a great time. No, not really. Socialization to other animals, to people and to environments is primarily achieved in the first few months of life.
Older dogs can sometimes benefit by repeated exposures to other dogs if introduced properly.
But it seems that the ones that have problems with other dogs, as you've explained, are separated and not allowed to join in any reindeer games. So a socialization program it's really not. Pack life involves greeting and affection, caring for one another and play, among other social behaviors. Dogs do hunt as packs, and when dogs are loose and running as a pack, the excitement of one fuels the excitement of the others, and so they get each other all excited. In this state, a show of aggression by one can invoke such a reaction in the others, and hence, a group or pack mentality.