Wouldn't it be possible to anesthetize the animals before killing them with some plants found locally ?
You know, most of the really good anesthetics from plants happen to be illegal...
On a marginally more serious note - you need to be careful, for exactly the same reason you need to be careful if you're eating slugs or snails from the wild - anything the animal eats before you eat it, is potentially still in its system to go into yours.
If you're using an implement with a blade, I think the main thing is that it's sharp - so that it cleanly severs nerve endings and causes less tearing of the tissue. You need to do it fast and accurately to keep pain and distress to the minimum. If you sever the carotoid (or presumably cause any major loss of blood pressure) I would think the animal would be unconscious (and hence feeling no pain) rapidly as with humans.
I'm not looking forward to killing our eventual chicken. It's even possible that my daughter (and then wife) won't allow it. For that reason rabbits are absolutely out of the question, and I wouldn't even know where to start with a pig.
While I'm not sure anyone should look forward to killing anything it gets more routine in short order.
However - there are quite a few things I think people who abhor/deplore killing animals to eat overlook. One is the question of if they eat meat - that has likely been killed in much worse circumstances in an industrial facility. Why should it be OK to eat meat where the pain and suffering is invisible to you and pretend that makes it alright, but not OK to do it personally as well as you can? There is a form of denial going on here - that makes it easier for out of sight out of mind suffering to occur.
Also, death is a natural part of life. Everything dies sometime - and I'm not at all sure it's less cruel to let an an animal die of old age or whatever condition (including wild animal) might otherwise kill it prematurely. I think parents do a disservice to their children when they tell lies about what happened to family pets that died - that they went away, went to puppy heaven, whatever - it's about as good a teachable moment for death as you could wish for, given that later their grandparents will go the same way (and later still their parents).
If I might make one comment about family animals and not making them hard to kill I would say - don't give them a name, not unless it's "breakfast", "lunch" or "dinner". Once named there is a tendency to build up an emotional connection (and some animals do genuinely have personality, not so sure about chickens in that way though).
The one animal that my parents wouldn't ever kill was a duck I rescued - it had problems hatching that had lasted so long they wrote it off and assumed it would die. I took it on, kept it warm, helped it a little to get out (which you're not really meant to do) and fed it etc and it made it. It got a name ("ducky", not very imaginative, now that I look back on it) and it was off limits for slaughter (as it was
my duck - a drake actually - and a fairly large one when it finished growing as it had a touch of Aylesbury).
With both chickens and ducks they can be productive egg layers for years before they start to get old and stew starts to make sense. If you're letting them mate (you don't want too many males - but you will need at least one for ducks which only lay fertilised eggs) you can always do meat and eggs simultaneously, though I get the impression you're talking a singular chicken - where eggs might make as much sense?