It's a tough topic to discuss. Most people of the SOS opinion have had first hand experience with them and their effects on personal safety and livelihood, making it hard to defend allowing the coyote to co-exist.
I have had no direct experience with them other than seeing them in the woods where I live. They're big NE coyotes but all they've done is run away from me or my dog. That said, knowing that they inhabit the area, My young kids do not play outside without supervision. I must admit that on this latter point I have greater fear of some sick human eyeing my kids than a coyote.
Taking all of the above in mind. I think that coyotes have a right to exist in
their habitat. The reason that there is an overpopulation in some areas is that before humans upset the balance, coyotes were kept in check by wolves. Wolves being top predators, are pack animals and possess larger territories, intrinsically maintain a lower density than coyotes do. They also tend to give a wider berth to humans than coyotes do, and will less readily mate with feral dogs. The removal of wolves from their original ranges facilitates an increase in numbers of coyotes which generally live in pairs and offspring vacate the family much sooner.
What we have now is an unnatural state where there is less territory to hunt and even the dynamic among coyotes is thrown off. The existence of bachelor packs is much more common than in the past. Nowadays, by killing coyotes on sight, and not fully understanding the particular population in a certain area, the risk is that an established pair which can keep younger bachelors out, can be eliminated, inviting a less balanced group in with greater food requirements.
Another factor is that coyotes are often blamed for the actions of feral dogs and coy-dogs. Yet another consequence of our encroachment.
I have pondered before the question of why as humanity developed, in some areas of the world the relationship between humans and their competition was one of hatred and fear while in other areas it was one of reverence and kinship. The only answer I can come up with is human population density. The unfortunate reality is that our planet is too small for us now and we do and will decide the fate of all other species. Well, except for germs, and mosquitoes, and black flies, oh and cockroaches. All these I kill on sight.

. Never spiders though.