rcmodel;
I would like to point out that if you had hit that coyote with the 30-06 in the shoulder area, it would have killed them just as dead as the .220 or .22-250. If you were using hand loads, it would be quite easy to reduce charges if so desired.
I agree with you on the need to be mindful of what is beyond your target. But you need to be just as careful with a 22-250 as you do a 30-06.
My father rather distrusted coyote hunters, and would run them off if he caught them attempting to hunt coyotes on our property, and considering that it was considered great fun by some of the locals to get all liquored up and pile into a pickup truck and go chasing through the pastures shooting anything that moved, I agreed with him. Too many would leave gates open, cut fence wires and run over calves. Not the best ambassadors for responsible hunting out there.
I have hunted coyotes, but never in that manner, and since I was doing predator control, usually at long range, I used a 30-06 with a scope. I found with great amusement, when we started raising Texas Longhorns instead of Herefords, that they were better at coyote control than I was.
I do believe in using an appropriate caliber for the target, but I didn't have a .223 rifle back then, and was using Dad's 30-06 and his ammunition, (I was doing it for him). I never had properly hit coyotes run off with a leg blown off, and I was using his ammunition for White tail deer, because it was what we had, some times I was even using old military FMJ rounds, so that Dad could reload them for deer season. I don't remember loosing too many coyotes due to them running off after having been hit. Flat out missing some, yes.
Shot placement is crucial, whether using a .22 LR or a .300 magnum, as is being sure of your target's background.