I have used the cleaver and also the cone. I won't go back to the block ever again, far to violent and there seemed no reason for it to be that way. We use a cone and slit one side of the throat. The most violent part is the death throwes, even the intiial cut seems barely noticed by an upside down barely concious bird.
I strongly recommend you cut BOTH sides of the neck, right down to the carotid artery. You have to go deep enough to get the carotid arteries, not just the jugular vein. The carotid arteries lie deeper than the jugular. Carotid arteries run up each side of the neck feeding the brain with oxygenated blood under pressure. If you cut only one, certainly the chicken will bleed out, but not as quickly. As it bleeds out one side, the other side is supplying the brain with fresh blood so it stays conscious a lot longer. If you cut both sides well, there should be an almost immediate massive drop in blood pressure in the head and the bird should lose consciousness almost immediately. You know you've cut the carotid artery because it spurts under pressure. A jugular will just drip.
I agree that the chicken barely seems to notice cutting the neck. I think the trick is to have the skin pulled really tight so you get a deep clean cut. I read an article written by a man who was stabbed in the neck and came very close to bleeding to death. He wrote about the experience and that he was surprised that he didn't feel the actual cut at all. He knew he was in big trouble because he saw all the blood, not because he felt himself being stabbed.
All the above said, I don't think they suffer much with the longer time of just cutting one carotid artery or even missing that and just getting the jugular. I don't think it is particularly painful, but I do think the chicken is somewhat aware and somewhat stressed. I want to shorten that time between initial cut and loss of consciousness.
I've never witnessed the chop method other than a YouTube video. While it looks violent, I can't believe it is for the chicken. I imagine that there is immediate loss of blood pressure to the brain and almost instant loss of consciousness. I think there is a fairly high risk that there will be mistakes made that cause tremendous pain and suffering to the bird by the inexperienced among us.
I'm going to have to read up on the broomstick method. I think that to do it right, you also need to cut the carotid arteries and bleed them out.. There are arteries that run up the spine that I am sure will be ruptured by cervical dislocation. I'm not sure that is enough of a bleed out. I once processed a Silkie who drowned in my pool, and the carcass didn't seem any different to the ones that I had bled out.
My fear is that I'll not use enough force and just dislocate the neck and not severe the spine. I have a bird or two, maybe three, that I need to dispatch. One is a 9-month-old really cockerel who is just too rude to the hens and is terrorizing some of them. He's a really big bird. The other is a 19-month-old aggressive rooster. He'll have really strong ligaments so I am a bit afraid to use a new method--the broomstick--with him although he is so nasty to me (not the hens, which is why he still lives--I need to replace him and so far all my cockerels have been jerks to the hens) that I might not feel too bad if I made a mistake. The third is a darling little Cochin hen that I just can't make healthy. She was a very expensive show bird bought from a breeder. I'm having trouble dispatching the hens I've developed relationships with. Ya, I'm a weenie.