Are you euthanizing a sick bird or slaughtering for the meat?
I prefer to use a cone and jugular/carotid slit for harvesting meat.
But use broomstick cervical dislocation for euthanasia.
This is the best vid I've found on the broomstick method,
the only one that didn't also remove the head.
Cervical Dislocation is shown in this video at about 1:00,
These two things are the keys to success IMO.
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Notice the slight divot in the ground under the stick and neck, this will keep the bird from being choked.
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Notice that she slowly and fully stretches out the neck and legs before giving the short sharp jerk that breaks the neck close to the skull.
I've found this technique to be very effective.