As stated earlier, hunters do actually shoot them and let them lay and that is a waste of a great resource. Coyotes are a great animal and should be treated with respect as we should all animals. As far as the coyote hunting requiring a furbearer license, I would say yes to keep some of the hunters in deer season from using the outdoors.
As far as the animal exploding, I dont think that is the case yet. There are definitely more coyotes than 10 years ago but not the numbers needed for an all out war on the animal. I believe trappers and hunters are averaging about 10,000-12,000 coyotes taken per year in PA since the year 2000 and that is not an extreme amount considering other parts of nation and with our number of hunters and trappers.
Whether the hunting public respects them or not, "we should never take from nature but just take what nature gives us" and that means not leaving coyotes or any other animal laying in wooods to rot.
Alan