It sounds like you've been to trainers who don't know what they are doing.He "knows" commands but does not always follow them, or takes his sweet time doing so. Especially poor at recall and at times will still run from us if we are trying to fetch him from the property or when trying to load him in the car. He is not a dog that wants to please the humans, just himself.
From everything I have read about Great Pyrenees, it seems this is a common issue and he behaves much more like a Pyrenees than a GSD. Livestock Guardian dogs are designed to work independently from humans and think for themselves, which he definitely does. We have been to several trainers, both private and group, and work with him every day, but his need to roam, explore, and make his own decisions is super strong.
We have come to terms with it, and just ensure he is on leash when off our property. That being said, he is super sensitive and if he senses we are frustrated, or give him corrections, he avoids us even more, so we try to use positive only. My husband has given some serious corrections, and it did not help.
Is your suggestion that we correct him strongly when he snaps at her? Thanks for your reply.
The Husky part too