I've been browsing the backyard chickens forum for a while now, but just learned something I think might be useful to others so wanted to add it to this post.
I'll confess that I haven't read all 12 pages of this conversation, so if this is already posted, apologies.
My husband and I culled five of our chickens last Sunday and we found that bolt cutters (proper heavy duty ones) worked very well. There is no humane-type advantage of this over a skilfully-swung axe, however there was a safety advantage for us humans. Neither my husband nor myself was completely confident that our axe skills would ensure a quick, clean, safe kill, so I'd definitely recommend this for anyone who feels the same about their axe skills and doesn't want to cause unnecessary distress to their chickens with a failed axe swing. We were also able to hold the chickens (they've always been comfortable with handling) and keep them calm until the moment that the blades were in their necks, and of course less than a second later it was over. This way we also had no need to induce the catatonia by holding them upside-down (I've not read anything that specifically studied chickens, but catatonia induced by being held upside-down has been shown to coincide with extreme stress hormones in rabbits and catatonia in humans is often linked with stress-related disorders).
I hope this helps!