I will confess to being too squeamish to do my own culling, or butchering these days. My husband does it for me. I definitely admire that you can do it. It's a necessary need in chicken keeping. I helped as a kid with butchering the chickens, so I'm familiar with it all.I use the cone and jugular/carotid slit method when slaughtering cockerels and old hens for meat. Kills and drains at same time. Yes, you need a good sharp knife, a knowledge of the anatomy(which I found here), and hardest is avoiding cutting thru the feathers(they don't cut easily), you need to go mostly between them. All that takes some practice, I've had several 'bad cuts' including one of my own fingers, before getting better at it. You also need good cone, I built mine from flashing aluminum using dimensions from a commercial cone.
For euthanizing a sick bird I don't intend to eat, I use the broomstick cervical dislocation method. Some use this for meat harvesting too. I did a lot of research and finally found a vid that didn't remove the head. It ended up being easier than I thought and removed the 'bloody' aspect.
-Notice the slight divot in the ground under the stick and neck, this will keep the bird from being choked as you get them set up.
-Notice that she slowly stretches out the neck and legs before giving the short sharp jerk that breaks the neck close to the skull, this is key to success IMO.
No one method will be without some emotional trauma to the keeper, the first time I killed a bird I had the adrenaline shakes for a good 20 minutes, and all have possibilities of 'not going well'.
