I agree with most of what you say. The problem with coyotes is they adapt to suburban life so well, their numbers go unmitigated. They explode in population and there is no predation to remove them.
This is true in many areas but the food supply is what dictates their numbers increasing. The more food the bigger their litters and the better their survival rate. Too many people leave garbage out where they can reach it. Farmers and hunters dump offal where the coyotes can feed on it rather than bury or burn it. Ranchers don't remove and bury or burn dead cattle and calves. Others let their cats out to soon become coyote food, get another cat, repeat, repeat etc. We've always discouraged coyotes by using those snap-lock trash cans.
They consume their natural food sources, rabbits, bowles, etc., then they wipe out the desireables like deer, quail, turkeys, etc. When they are done with that, they start looking for animals like people's pets.and coming into your yards.
That's where good fencing comes in. Coyotes have never been known to drive any other creature into extinction. When their food supply runs low - not out, coyote (or wolf or cougar) numbers start to drop. But yes, they will, in desperation, enter towns where people leave garbage and edibles for them to find. Alleyways behind food stores and restaurants with trash piled up is going to attract hungry predators as will those dumpsters full of food behind Malls and large grocery stores. This a people problem as well as a coyote problem.
The 1st 10 years I owned my property we rarely heard coyotes and never saw one. Within a couple years span, they did exactly as I described. I started out firing over their heads with birdshot. Then I migrated to firing at them with a 12 ga but usually at 50 yards or better. Then one day I witnessed them repeatedly trying to lure my male airedale out of bounds of his invisible fence. One would come inside the boundry and provoke a chase, while 2 - 3 others were waiting out about 200 yards. That is when I declared war.
Why are you using an invisible fence where you know there are coyotes? This is what I mean about it being a people problem as well as a coyote problem.
Rather than kill them, and you can't kill them all, it only takes one to kill your dog (if that was their motive), put up a good fence as I did. The coyotes here would run behind the fence and look at my goats and chickens but they knew running along the top of the field fence was a string of electric wire. No way they were going to get over the fence for an easy meal. In all those years, I never lost one goat to the coyotes.
And the same fence was used for our three dogs. I bought a 330' roll at the Farmers Co-op. We never lost one of them to the coyotes either.