Interesting that you and other comments took mine to mean how to train a dog, which is irrelevant to the situation under consideration. Did any of y'all mean that in good faith?
No, it's about how to de-escalate a severe situation without loss of life. (Good) policemen don't break up a fight between an 18 yr old and 10 yr old by viciously beating and then stabbing the 18 yr old. My point was only that the instincts of those involved made the situation worse before it was made better. And based on the new facts, starting with the owner.
Just as with humans, there's a well understood hierarchy of severity of tools for handling situations such as that one, though only a small minority of dog owners know them. It's incredibly unfortunate that dog lives are more or less toys for humans to play with, otherwise this fight is likely to never have happened if it was a severe as you say before it turned into a five animal brawl.
Bottom line the three of you DID make it worse before it was made better. The punching and kicking won't help in all probability. It's simply not on the spectrum of effective tools. Killing is on the far end, so I'm not begrudging your actions, I'm begrudging the behavior before it: yelling (probably), kicking, and punching instead of forcing the bite, wheelbarrowing, and other environmental tools that may have been available. Y'know, the things that animal behaviorists have figured out, how to cause specific behaviors, like stopping a dog from biting another during a fight.
I'm not sure why I wrote this comment since apparently your only regret is not achieving your intent which was to kill the dog. You managed to stop the fight without causing death, a huge win, but you wish otherwise. I sincerely hope your future is brighter than your past.