I looked into LG dogs when I first moved here. It only really works if
- Your land is ring-fenced
- No roads cross your land
- No public footpaths cross your land
- (Presumeably) Your land is not what we in England call 'Access Land', ie., where members of the public have a right to roam (and isn't this all of Scotland?)
Unless you can enlighten me more, FW - I talked to people with LG dogs in other countries, didn't manage to discover that we had local expertise!
I was given to understand that the dogs would guard the livestock against all 'invaders'; that they learn who is 'family' but would not get the hang of random walkers crossing meadows on footpaths. And I was certainly given the impression that they might do more than bark 
From what I've heard of guardian dogs in the US, they have to live with the flock from when they're tiny puppies, so they think they're sheep. This way, wherever the flock is, so is the dog, so fencing isn't an issue with a hefted flock.
The one we saw in the Auvergne wandered across to check us out, but there was a ratty fence along the roadside (it could easily have hopped over had it wanted to). This was in the mountains, and I couldn't see any other fences.
But yes, I get your point that British access laws would mean a guardian dog could be in big trouble. Conna did once bound across a field when some people arrived, who she didn't know, and they walked straight onto our land unannounced. She barked furiously and the woman was livid, but Conna didn't attack, just gave them one ginormous fright

. You couldn't trust that it wouldn't happen though. Our set-up here, with double fencing and hedges the whole way round only leaves the gates for critturs to jump over, or sheep mesh fence for small ones to push through. We do have a road running through our place, but we keep only tups on the other side.
In the context of Lynx, I can't say how a guardian dog would perform. Would it be able to frighten the lynx away, or would it have to fight? Cat v dog, both big and strong.... I've always felt that guardians would be better having two to look after each flock, especially where there are wolves, bears etc.