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I'm afraid you are quoting a persistent urban myth. Rest assured that your state DNR/Game & Fish, does not release apex predators into an urban environment. I have talked to them about this, they laughed and said "Yeah, and we tie damsels in distress to the railroad tracks too!".
The fact is some states have a year round "season" without bag limits in a somewhat futile effort to control them. Getting rid of coyotes is about as easy a getting rid of raccoons. You may get a bunch, but they breed so fast that they will be back soon. As others have stated coyotes do very well living amongst man. While Coyotes will prey on deer (mostly fawns) the biggest threat is to small game, which of course is why our birds are once again on the menu. As we expand our "territory" more and more, we will have more and more problems.
Sounds like you did about as much as you could, short of building your run out of chain link/sheep panel/stone. I have a very similar setup to yours from the sound of it. Some times you just get unlucky. That doesn't help how you feel when you loose your birds though. In the last week I have relocated 2 possums, and 2 raccoons (just came in from skinning the latest) to the bottom of my gator hole. Probably not even a dent in the population but I'm getting rid of the bolder ones, that's all I can do. I'll set the trap again tonight. The coyotes have left me alone so far, but we have plenty here.
Swamp
I didn't say that I believed it but I was told this by a DNR/ fish and game agent who came to my place to investigate a report that I was keeping and abusing a Canada goose ( one of the city people saw me using a herding stick to keep my african away from my mower while I was mowing). Coyote are native to the mountains to the west and the swamps to the east in NC. also not apex preditors since we still have bears, wolf. and big cats called 'panthers' running in the less populated areas of the state. Due to my camera system I have seen coons and possums trying out my fence and leaving after getting zaped. BTW we also have problems with peafowl excaping to the wild and competing with wild turkey for food.