Nowhere is "humane" defined as being easier on the one doing the killing. I think that's a terrible way to think about it, any good animal steward should dismiss that definition. It's hardly how vets, rescues, or even laboratories define it. What matters is the animal's wellbeing, not the dispatchers. Frankly, it's not the dispatcher's life that's about to end. If someone tosses a bag of kittens in a river they never have to think about them again, it might be SUPER easy for a person to do that (as is evidenced by the fact that this is actually depressingly common) and many people feel very mentally sound doing this, but it's terribly cruel to the kittens. That's why It's illegal. So many people thought it was the easiest thing to do that we had to pass laws against it and it STILL happens.
The American Veterinary Medical Association defines it as;
"Euthanasia is derived from the Greek terms eu meaning good and thanatos meaning death. The term is usually used to describe ending the life of an individual animal in a way that minimizes or eliminates pain and distress. A good death is tantamount to the humane termination of an animal’s life. In the context of these Guidelines, the veterinarian’s prima facie duty in carrying out euthanasia includes, but is not limited to, (1) his or her humane disposition to induce death in a manner that is in accord with an animal’s interest and/or because it is a matter of welfare, and (2) the use of humane techniques to induce the most rapid and painless and distress-free death possible. These conditions, while separate, are not mutually exclusive and are codependent."
I also think it's incorrect to suggest that slaughter and euthanasia are totally unrelated. especially for back yard operations. I think the whole reason many of us raise our own food is because we can't stand the way that commercial industries handle animals BECAUSE it is inhumane. We seek out the most humane options for our animals because we raise them by hand, we live with them, we take care of them every day of their lives, we care about them immensely, and we want what's best for them, even in death. If you can't believe that, then I think it's safe to believe that at least we try for something that doesn't distress the animal because adrenaline causes the meat to stiffen and become acidic. We genuinely want the fastest, least distressing method when we dispatch our birds. And frankly, if gassing were that option.... Well, baking soda and vinegar are cheap, it would add pennies onto my cost per bird. I would gas them every time. I just genuinely think cervical dislocation is more humane.
Unfortunately, not every situation gives the opportunity to see a vet. Vets can be expensive, closed or even just unwilling to work with chickens. I, personally, think that for a pet the vet is ALWAYS the best first option (It's peaceful for everyone, respectful, and extremely humane for the animal) but there are occasions when that's just not an option you can take. And having a back up plan you can do yourself is important for those occasions to prevent the suffering of the animal.