Hi John, and welcome to permies!
Slitting the throat, when done quickly and cleanly, is very humane. The key is slicing through the carotid artery. This causes an immediate loss of blood to the brain, and unconsciousness happens within seconds. It looks a lot worse than it is, because of all the blood and the way that the animal will thrash and kick when systems are shutting down.
The problem with shooting, according to my homekill butcher, is that you don't have a very big target on a goat (or sheep, alpaca or other medium-size livestock), and there's a greater risk of wounding without stunning. Here most cattle are shot with a .22 and this stuns them so that the throat can be cut, but sheep and goats just get the knife.