The collar fight was almost certainly the cause of the cough, but not much point castigating yourself over it --- some animals go berserk, some won't, and you can't guarantee which will or won't. Generally I've found dogs to be more likely to freak out about collars, leashes etc than sheep or goats.
Chances are he'll be fine in time, but good to be aware of the possibility of them strangling themselves even without anything to wind around other than their own legs or if there's no swivel on the chain/cable, which is a big risk.
I've leash/chain trained goats and sheep before and they're generally very easy to train but I most likely would not allow one to work itself into a frenzy, though to be fair that's hard to prevent if that animal is so inclined. Good to jump in and put a stop to it as soon as is safe and possible, generally speaking. Also they'll trust you more so in future if they get tangled and hurt in a fence or whatever they're more likely to listen to you, stop thrashing, stay still and let you help them, which can be the difference between life or death for them. Good can come of bad situations.
It's a better idea to allow them to drag a leash around for a while, not anchored to anything, so they get used to the pull, and the tripping obstacle, and learn to manage it before being restrained via that method. But be aware some may panic and head for the hills, and when in a panic some sheep and goats will leap directly into walls or obstacles they should not, or try to gallop through fences at top speed.
I don't have sheep fencing up yet on the new place I've moved to so my pet sheep are on cables or chains and while they're very sensible and smart and never had that epic battle you're describing, my ewe does have the unfortunate habit of leaning all her weight into the collar so she's walking on an angle to try to meet me in the yard; she's broken collars that way before and once they learn, they will usually repeat the behavior.
Because your little fellow had what he may have perceived as a life or death fight which he won after not surrendering, he will almost certainly repeat that massive overreaction and in doing so only confirm his initial finding --- this is how you manage to escape this situation. Goats in particular need very good restraints, fencing etc so they learn to not challenge it. Most sheep are the same. If they get around, under, through, or over a fence or escape or break a collar or chain just once that can often be all they need to repeat that trick like a professional. They're not stupid, not even sheep despite the common stereotype, they're quite smart on average.
You may have to get a panel yard for him or invest in some dog mesh fencing or chainlink fencing or something like that rather than trying to chain him again, but as a pet it's likely he will need to be on a collar or leash for some reason or another sooner or later, so it may be a good idea to put a sturdy collar on him and a weighted leash (not too much weight, just enough weight that he knows he's dragging it around, so like a thick rope maybe except of course if you use real rope he'll eat it), and then just let him drag that around for a day or more before trying to get him used to leading, then trying to tether except with treats or something to help occupy him... Just suggestions, may or may not work.
I had one little goat I had to chain because she would roam too far, alone, too adroit at climbing grapevines and too hard to contain, and she was only a baby but broke chains of many grades like they were soft, but you could pull cars with those chains or chain other animals... (We actually did, too). She just had breaking them down to a fine art somehow. Rapidly she understood that the flat-out gallop which ended in her landing on her back would break the welded links. So that's what she did. She'd be grazing peacefully one moment then bolt for it, then get up and graze, then bolt for it, until she broke the chain. She went through dozens of chains, would have been about 6 months old at that point and was still physically quite petite. Wire link chains twisted together are better than solid, welded links as they have some give in them so don't just snap.
Best wishes.
