You can use tethers for goats, yes they have to be fairly stout so the goat won't chew through it quickly but if they have lots of tasty browse around, they are far less likely to think about chewing the tether rope.
The trick is to move them to a spot in the morning and then put them back in the one high fenced area that also has their house inside, with some fresh
hay to eat, straw for bedding and good supply of fresh
water, they will be contented goats.
Doing a controlled burn (as you mentioned) is another way, but you aren't going to get the good manure, trampling, etc. that you get with animals. You will end up with a bit of bio char though and that isn't a bad thing, the quantity would not be great though.
Catalpa trees are or can be a money maker, the catalpa worm is a highly sought after bait for many fishermen. Collect them into paper sacks and 50 go for around 4 dollars here. (catfish can not pass up a hook baited with a catalpa worm nor can bream, black bass or perch.
If the "scrub" pecan doesn't produce good quality nuts, it does have pecan
wood, highly prized by furniture makers all over.
Remember that the wild plants you have mentioned do have value if for no other reason than being a source of mycorrhizal fungi both in and wrapped tightly around their
root systems.
You don't have to limit yourself to goats, donkey's and hogs are also great animals for clearing
land of excess vegetation. (our donkey eats most of the day, while she likes to graze on grasses, she also loves to prune trees and the end shoots of brambles, blackberries, wild grape vines are all eaten with great relish, like they were beluga caviar on brioche.
These are just the things that popped into my head as I read your post.
Redhawk