Like John, I'm skeptical of this method as well. It's very much a commercialized
production approach, rather than a
permaculture approach. You won't find it in any natural ecosystem. You've already spotted the pitfalls, which create a lot of work to avoid.
I have goats, which are free to graze (rotationally), but are in their
shelters at night with free choice
hay. I use the deep litter method in the barn and find it works very well. My bedding is wasted hay, mostly. I rarely buy straw. Several times a year I clean out the barn and top dress the pastures with it. I especially search out bare spots, where I toss down a forage seed mix and cover it with the barn litter. For me, this is the simplest and easiest workflow. My goats are healthy and happy, and my pastures are improving.
I do gather branches and weeds for the goats, but it's more of a treat. I agree it would be a lot of work to gather enough to keep them in good condition. Remember too, that goats, sheep, and cows are ruminants. They require long-stemmed grasses and hay to stay in top health because that's how their digestive systems are designed to work.