Its like hanging. If you do it properly, the neck snaps and the victim dies instantly. If you do it wrong, the victim dies of asphyxiation, and its not.
In that respect, its no different than any other killing method. Quick and certain is much more humane to the victim than slow, halting, and/or clumsy.
I use a SHARP! knike. Not merely Sharp, or sort of sHArP, or was once sharp, but actually SHARP!. Take the whole head off in one stroke (Its my favorite 8"chef's knife). The mental fortitude to do so, and the clarity of mind to work quickly on those rare occasions when the knife doesn't glide between vertebrae, is not found in every individual.
Important that owners find a humane method that works for them - and if they can't bring themselves to do so with certainty and quickly developed skill, then in my opinion, they should reconsider whether or no they can be responsible owners.
"When the need arises - and it does - you must be able to shoot your own dog. Don't farm it out — that doesn't make it nicer, it makes it worse."
Robert A. Heinlein
Your chickens, too.
and yes, a headless chicken will flap around quite a lot - autonomic nervous system doing what it does. Headless snakes behave the same way. So does a lizard's tail. All quite similar, from an evolutionary perspective. Its not a sign of "life", don't dwell on it.