We have a livestock guarding breed also.
Yes, LGD are very independent minded and do not work to command, that is true. They are not trainable and rarely have reliable recall. I agree that not all of them are cuddly, though some are. However, unless the dog is kept outside with livestock, the family becomes the flock and without them, the dog is lost. So it is not true to say that LGD are not prone to SA.
Some LGD suffer terribly when they are left alone. We've had 9 of them, all as house pets, all reared by us from 8 - 10 weeks old. A couple of them were as you describe, not fussed about being left on their own, the others had some degree of separation anxiety and would whine and bark. Three of them lived as a multi dog household so obviously that had some bearing.
Our current girl is a lone dog and is not too bad about being left for short periods. She also sleeps downstairs and often chooses to be in a different room from us. She is not clingy and not a cuddler. Our previous boy was also a lone dog and was the total opposite - cuddly, clingy, hated being left alone, would bark and whine. We literally could not go anywhere without him.
Both the same breed, both from working bloodlines, reared by us from 8 weeks old, same house, same routine. Totally different characters.
So, in our experience, the level of SA varies greatly from one dog to another. It is not breed specific. They are all individuals.
I definitely would not recommend that anybody take on a gigantic guarding breed with the idea that it will be so independent minded that it won't have separation anxiety. Because, in our experience at least, it's more down to the individual dog, not the breed or type.