Here are some links that discuss "genetic drift." I'm not a biologist, and this reply is my first attempt to understand this concept. This reply is a sort of written "thinking aloud."
http://en.wikipedia.org/wiki/Genetic_drift
http://evolution.berkeley.edu/evosite/evo101/IIIDGeneticdrift.shtml
http://www.britannica.com/eb/article-9036394/genetic-drift
I'm not sure what you are describing can be attributed to genetic drift. It seems unlikely it is a result of genetic drift resulting from some "founder effect" like one of the many genetic condition common in modern specific dog breeds, although this too is possible if the dog were a pure breed (a genetic aggression problem was a major issue for St. Bernard breeders some years ago).
There seem to be conflicting views on how the process of domestication (the original evolution of the "village dog" from wild canids) affects canine aggression, and the more recent concern of how modern selective breeding affects canine aggression.
I do not think it is at all clear that dogs are less aggressive than their wolf-like progenitors. It must depend on definition of aggression, breed, etc. wolves may have more "hunt" and "prey" drive, but this is a very different form of aggression than aggression between pack-mates, for example, or against humans.
Today's third world village dogs are still keen competitors for resources, and therefore capable of limited aggression especially with each other. My wife met a very bold street dog in Athens very intent on food in her back pack, showing that some will try to take resources by force from humans: much like a ball crazy ACD or GSD. Therefore aggression is no doubt a part of "natural selection" of dogs which is the opposite of qualities selected by genetic drift.
Conversely, village dogs that show aggression toward humans, and prey motivated aggression toward domestic prey species (chickens, ducks, sheep) need to have a mechanism to inhibit prey motivated aggression or inappropriate aggression toward humans, and no doubt this is naturally selected for as well. Raymond and Laura Coppinger in their book "Dogs" discuss this inhibitory effect at length, including explaining how important early conditioning is to this inhibitory process. It would seem that this sort of inhibition of prey drive is much more important for domestic dogs than for wild canids.
Also, village dogs no doubt "avoided" a lot of conflict. I'm certain that guardian breeds are selected to NOT avoid conflict, although in fact most still do avoid most conflicts, if possible.
I'd also note that wolves to function in a pack also need to have inhibitory elements to aggression. It is also clear that pure wolves are far less aggressive toward humans than dogs as a rule.
All I would conclude is that in wolves, "village dogs" and modern selectively bred dogs that there is a tension between both aggression and prey drives, and the inhibitions necessary for survival and success. Some of this is genetic, much depends on conditioning. Therefore, aggression problems are likely to always be with us.
I don't think there will ever be a population of dogs that has no aggression, in the way certain populations of humans lack certain blood types as a result of founder effect and genetic drift.
So my answer to your question: the problem you describe, I bet has nothing to do with genetic drift.
rgds, Andrew May