If at present you can't get within 20ft of them, then even with kind words and bribes of favourite food, it will, as you say, take a long time to get their confidence. In fact, as distust is their default setting, you may never achieve it.
Goats are prey animals. These ones obviously see humans as predators. So in my book, you'll have to do something more positive to convince them otherwise.
So - before you go any further - stop looking them in the eye when you approach. In fact try not to look them in the eye at all until you have their confidence. Eying the prey is what predators do when moving in for the kill, so when you do it then you are just re enforcing the goats' worse fears. Look at the ground or rear of the animal. You can still see what they're doing but they stay calmer.
You need to herd or somehow trap the animals in a pen or shed from which they can't escape. Then get a collar or halter on them and tie them up to something strong. Make sure you can easily release them if need be and don't leave them alone and tied up.
Once tied up, they'll probably jump up and down frantically for a few minutes or throw themselves on the ground. Just give them some space and they'll soon calm down.
Keeping your distance, get a long branch, or long stick with a glove on the end and stroke the animal gently. After a while they'll realise that this contact doesn't kill them, and in fact doesn't hurt. It might even be pleasant. At this stage I don't go near enough to invade the animal's personal space. (Hence the long stick/branch) But keep moving about so it gets used to my presence without feeling too threatened.
When it remains calm when stroked from afar and has stopped panicking as you move about, then you can approach it. I usually then use my hand to gently and briefly scratch its backside. Once it's realised that this is quite pleasant, you can move your hand to gently scratch or stroke it's back and then move gradually to its shoulders. When it appears relaxed at you touching it, then give it some treat to reward it and let it go.
Then do the same the next day until it is happy for you to touch it without being restrained.
If the animal is that wild that you don't get as far as hand stroking in one go, then just get it used to being stroked from afar, give it a treat and release it. Next day it should at least be a bit less worried, so try and progress a bit further.
Good luck.