There are some different thoughts to the "good coyotes - bad coyotes". Has to do with the establishment of their range and that some are more inclined to go after larger prey then others. The concensus is if they are not bothering the cows/sheep/chickens, then don't kill them as they have established a dominance and that they will keep others out. For 2 years there was a pair at one place and they were seen often digging for mice and small game. Never bothered the cows with small calves, and the cows just ignored them in the field. Then some neighbors decided they "had to kill them" because they MIGHT bother their chickens, and someone told them that "all coyotes were killers and they needed to stop the problem before it started.
Well, they did kill one, the male I think; then the other was seen fighting with a new one (different coloration so we knew it was new to the area) and not 2 weeks later the cows were all in an uproar and the people started losing chickens. We finally wound up shooting one that had taken to running the new calves, and had the cows in a nervous state all the time. So there is something to it, but you don't know if you get the "good ones" when they show up. But we have been careful to watch and if they aren't causing a problem, we leave well enough alone like that pair. If we have a problem, then it is war on whatever ones we see.
Have some here that I hear about every 2-3 weeks, but they don't come down to bother the chickens in their "moveable" pens, so I leave well enough alone. There are a large number of barn cats here and I haven't noticed any thinning of the ranks, so they must not be bothering them either.