Until I retired from agriculture, I raised sheep in upstate New York, within sight of the Canadian border. We had big, eastern coyotes who could become fairly aggressive. On my farm, coyote control wasn't hunting; it was war. I have had coyotes look me in the eye at 30 to 40 yards--and then continue an attack on a sheep, or my border collie. It was rare, but it happened--and was reported to the state, where it was classified as an "incident" only one level below an attack on a human being.
Coyotes were engaged at every opportunity, with any weapon at hand. Once, in defense of my working dog, and unarmed, I simply ran at a big female and backed her off. (That was the reported incident.) I shot her the next day, (she had patterned us). She weighed 55 pounds. When I told a State biologist conducting a seminar about livestock predation, about my incident, and how it was resolved, he grinned and said, "I love North Country people".
For coyotes, I greatly preferred my .270 to my recurve. I can relate to your adrenaline....
I can appreciate your desire for a quick, clean, kill. I always tried for that, as well, but in defense of pets or livestock, I took the shot I was given, with whatever I had. It sounds bad, I know, but a grievious wound was as good as a kill, for my purposes. When they are packed-up, I do not believe coyotes look upon an injured member with great sympathy....
I fully agree with the advice you have been given, I would only add: if you can get a hit, get the hit. You are right, "this isn't about fair play, anymore". Take the shot. Get the hit.
Good luck; good hunting.