Coyotes, and ...others
Your problem with people who just seem to take pleasure in killing animals is justified. Sportsmen have had this problem for a long time. We used to call them "city hunters", and other less savory names.
See, the main problem is that anyone who buys a gun and steps off the pavement is a "hunter". NO matter how well or how poorly they exibit sportsmanship, to the press and the non-hunting public, they are "hunters", and the worst people tend to become the image that is held up to the public. The entertainment industry is dead set against what they call "blood sports" (unless it involves men in a ring that they can make money from), and killing animals is "cruel". These people have never seen, and never will see nature in it's actuality, only in the edited version you get on TV. It is not more cruel to humanely kill any animal with a gun, bow, or even traps, than it is for them to die a "natural" death. A natural death for a wild animal is to be killed and eaten by a predator (not exactly painless), to die of disease, or to starve to death when old age cripples them to the point where they can't get food. Or to die of cold or starvation in the winter. All kind gentle painless gun things, right? Old animals don't spend their time being cared for in nursing homes until one day the pass away in their sleep. Harvesting game animals could actually be looked on as a kindness, and true sportsmen do their best to take game cleanly, humanely, and as painlessly as possible.
Each generation for the last few, fewer and fewer people have taken un sport hunting, and have fewer and fewer places and opportunities to hunt. Many previously hunted animals are now protected, not because their populations are in danger of extinction, but because hunting is "wrong", and many places formerly open to hunting are now closed to it.
Coyotes back east (I grew up in northern NY, so I do know a bit about it), have filled the role of general predator, and as you have noted, have few natural enemies, and man in that area is not doing his job of being one of them. The reason the coyote comes into your subdivision, shows no fear, and raids your garbage and eats your pets is that he is not being hunted! So he stands there, and looks at you, because you are not dangerous to him. If you started hunting them, you would find out pretty quick that once they realize you are dangerous to them (and it won't take long), that they won't just stand there so they can be shot "like fish in a barrel".
True, coyotes don't attack humans often (or even frequently), but that is because they eat everything else. Humans aren't worth the trouble.
Population cycles are just that. Cycles. Have a mild winter, more rabbits survive, next year, more coyotes. Too many coyotes, eat too many rabbits, fewer rabbits, then fewer coyotes. Up and down. We killed or drove off the big predators, and we used to kill enough of the smaller ones (coyote, fox, bobcat, etc), but we had a lot more people (hunters and ranchers/farmers) who were doing it. And we used traps, And we used poison. And we never really did more than hold our own in local areas.
Today, you can't use poison (never really cared for that one myself, too indiscriminate), most places you can't use traps that kill, and many places no traps at all, and there aren't nearly as many people who can take the time to hunt them. AND, you don't get hardly nothing from hunting them anymore, either. 30-40 years ago, coyote pelts could bring as much as $100 for a prime winter pelt. Made the time hunting them worth it and more.
Can't wear (real) fur today they say it is "wrong". No market for coyote pelts anymore, can't hardly even give them away. Why bother hunting them if you aren't a rancher. And if you are, you don't really have the time.
No matter what you see on Disney, they aren't nice critters. They can (and do) carry rabies. And one kind of rabies has NO visual symptoms. They can be rabid without the wild foaming at the mouth behavior. And if you get bit, you got it too! And rabies is fatal, eventually. We have treatment, but it is no fun at all.
No matter who you blame for how we got here (over hunting predators, habitat destruction, whatever) the fact is we are here, and we are not being allowed to do what needs to be done, as well as it needs to be done. Not enough people hunt, hunting is looked down on by the social elite, trapping and poison are out, and the same social elite won't spend the money needed for game depts to manage the animals. In fact, the majority of the money that state game depts do get comes from hunters and fishermen, through license fees and taxes on sporting goods.
Out where I now live, we used to be able to hunt cougars. And they were hunted with dogs. A handful of cougars were killed each season. About half a dozen years ago, they banned dogs from being used. You can still hunt cougars, just not with dogs. Today, almost no cougars are taken by hunters each season, as cats are really hard to find and track down without something like dogs. Several outfitters who used to make money for the state (through fees) got out of the business, because, without dogs, there isn't any business. Cougar population is way up, and they are coming into towns now, because they are no longer afraid of dogs (and men). In fact the state, instead of making money off the hunting, now has to spend money ($2 million last year) hiring out of state hunters and using game dept personel to hunt, trap, and kill nuiance cougars. Children have been menaced, and there have even been a couple of attacks, fortuanately non fatal.
This is the animal protectionist idea of proper animal management. Do nothing, nature will take care of it all. Sure, it will all work out, but as it does we have to listen to the people who caused it whine.