Not to detract this tread. But here is a question since you seem to know your dog behaviour stuff. My great peareneas lab mix is generally great and getting better with age( almost 2). He charges people(pedestrian or cyclist) passing down our mostly quiet road. Very difficult to call him off. I just don't know how to stop it altogether. Part of the reason for getting him is security.
I'm not a trainer by any stretch of the imagination, but a Great Pyrenees dog is a herd guardian breed, same as my sisters C.O. Wary by nature, a little more suspicious than most breeds, he has a job to do, and unless you provide one for him he will probably define his own job. You may be viewed as his herd, the people approaching your property he may view as interlopers and is doing his job to drive them away.
Dogs need training and interaction as well as stimulation, both mental and physical. A lot of people may view a big backyard as adequate exercise for their dogs, and for some it may be, but most need more. Every day my dogs get a walk through our woods. It is part of their routine and they probably view that as their job. New scents, different scenery, exercise, they all help.
Our dogs, while going through their "puppy" years, averaged 4-6 miles of dedicated exercise a day, sometimes getting as much as 10. High energy dogs require a LOT of exercise, and when denied that exercise they will find ways to try to drain that energy. Guess how much stuff got torn up in our house? None. When I left for work they were kenneled, and I had trained them that the kennel was a very good place to spend a few hours a day. They would run for their kennels, couldn't wait to get in them.
Most people on this site do a lot of research before they buy their tractor, or whatever it is they are buying, trying to make an informed purchase. The same degree of research should be put into a dog breed and breeder before making that purchase. Our one dog is a rescue and we got lucky with him. Our other dog came from parents that the owners bought from a good breeder in Florida (she was free, a gift from a friend after my old boxer died from cancer). Do you think I called that breeder to find out all I could about their breeding lines? Yes.