I struggle with this topic.. Because I find I can't even read the reasons why NOT to kill them.
I shoot every damn Coyote I have seen the past 40 yrs! I hunt them by calling them in.
The only coyote sitting under my tree is a dead one.
Your first line said it all " did not see any deer" well thats because the number one natural predator was sitting under your tree. He probably killed every fawn within 5 miles of your tree.. and the adult deer know what he smells like.
I could write about countless stories about the damage of coyotes to livestock on my father in laws farm!!
Or about how the wolves have wiped out the moose in alaska, and the deer in minnesota..
Maybe I would have a better chance if I told you to eat them... I heard they taste like chicken