I think you're confusing all conservationists and those who respect an animal's rights with those rabidly anti-human-rights people who often do more harm than good for animals and the environment through their activism. I support stewardship of the planet but not at the expense of human rights; predators are often cruel, that's just the way it is.
As for whether or not it's 'humane' for a coyote to rip a live chicken apart, no, because it's an animal, so it's under no onus to be 'humane' to another animal.
It's a mother training her young to hunt and kill so they can survive; no it's not pretty, nor kind, nor something I think you should put up with, but it's 'nature' as they say. By all means shoot her and her babies, they ought not to be foolish enough to hunt a human's territory or livestock; but don't be cruel to a creature because you don't like its cruelty to its prey. I do appreciate however how aggravating it can be to see the mess a predator makes of your pets or livestock, and I totally support your need to kill the predators responsible. I'm not trying to condemn you, so please don't take it that way, I'm just explaining my stance on the issue which afflicts basically all livestock owners. An animal's cruelty does not provide justification for ours, difficult though it may be sometimes. Clean and swift kills, with minimum suffering, I believe to be a human responsibility to the animals we share the world with that we choose to stop sharing our immediate environment with, lol.
I use cats for rodent control and often intervene to make sure the rodents are out of their misery in short order, but won't consider using bait that gives them horrible suffering that's worse than what my cats do. Cats can be cruel, but I've watched bait kill rodents, and it's slower by far and just grotesque. I can't risk it harming my livestock or other native animals either, so while modern baits are often quicker I would not consider them either. On that note I don't continue to employ cats who are maimers but not killers. They have a job to do, a cat that makes a mess but doesn't fix it is just making more work for me.