Another problem with tethering goats is they are not protected from dogs or any other predator that comes along. A
fence may not stop all dogs either, but it would at least slow one down and the goat could try to run away. If it's tethered it really doesn't have any chance of escape.
Solar electric fence will definitely discourage dogs.
Goats tethered in brush are likely to tangle themselves up in the rope in the first five minutes and (even if they don't hang themselves) you'll spend all your time untangling them.
I agree with other folks, the fence is a better option. Safer for the goats, a bit of up-front work in setting it up but far less hassle in the long run for you.