I used to live in GA and kept goats on 2 different farms for 20+ years. Worms are the big issue. Sometimes they like grass and they will eat it even when there is browse available, but the worms are a lot worse if they are forced to eat grass and other low stuff. In that case you
should rotate and use wormers. I had good luck with some herbal wormers ("Restore" is one brand I recall.....mostly wormwood....stands to reason) and occasionally resorted to ivermectin. The second homestead was much more woods and brush and the worms weren't much of a problem. But I still had to trim hooves. Goats are desert and mountain critters by nature and their hooves are being worn down (continuously, not occasionally) by sand and rock.....so I doubt you'll ever be able to do without that, at least on your breeders. (if you kill young goats at a year or so old for meat, these might not need trimming).
I found that cutting branches and hauling (I did this in winter from evergreen things....even loading my car up when I went to town!) and various forms of coppicing worked well. In the young woods I would whack a sapling halfway through and bend it over so they could reach the top....and later could reach the sprouts from the base. Any vines of honeysuckle, grape, kudzu, etc. are favorites too and can be dragged down out of the
trees with a hook on a pole. If you cut prunings, stand them up in circles of fencing or hang them in bundles....don't make them eat it off the ground if you can....this spread worms, which are the main pest of goats in damp climates....