First, sorry to hear you are losing birds. If there is a Murphy's Law on chickens, it may be when a predator attacks, they get your favorite first.
Second, on free ranging........I have concluded a true free ranging flock is not that much different than sending a hormone laden teenager off to an unsupervised, all night bikini beach party at an STD clinic. What could possibly go wrong? An extension of both is that unless you, the one responsible for their well being, are willing to accept the consequences, you need to put your foot down and say no.
So a reasonable solution to the lost bird problem is to establish a safe zone of protection for them. An enclosed run is one option. A larger yard is another. But as the size goes up, so does the cost of building it. The most cost effective way to provide a large are area of protection is to use an electric fence designed to keep poultry in and predators out. You can make this as large as you are willing to maintain, but make it easy on yourself. An acre can easily support as many as 50 birds, so that might be a good place to start. The key to keeping the birds within this zone of protection, is to provide them with a variety of food and cover options so they will accept what they get and not try to push it as far as expanding their range.
And properly built and maintained electric fences do work well to keep predators at bay. That is by day........the coop protects them at night.
BTW, as you describe it, a fox is high on the list of suspects. They can snatch and grab a bird without a trace. They take it home to feed the kids today and come back for another tomorrow. They keep coming back until all are gone and keep coming back long after that just in case you were kind enough to restock the cupboard.
On your LGD notion, if you have regular visitors to your property, you may have issues with some LGD's. I've never had one, but I've been around properties and livestock they were protecting and they were not friendly. But that is their nature and job. Seems a dog like that may cause trouble for you?