Cutting the throat seems to be the most painless, as any animal falls asleep from blood loss, but then it is messy. I have not had a cut on the throat but have suffered once when younger from a stab wound to the artery and yes you do fall asleep very quickly. I was lucky that someone was there to stop the bleeding until a ambulance arrived. I also suffer from allergic reactions, one of the forms of this is a rapid drop in blood pressure. The last time this happened was 3 years ago and I was in the doctors office, and after recovery told I was dead by the doctor momentarily. I passed out but could here the voices of the people working on me. Though they were very frantic I on the other hand was never so peaceful in my life. So blood loss to the brain seems like the most peaceful way to go.
When I process my chickens I will cut the throat. It is said that until the blood loss takes over that when people have been beheaded that they gasp for air. Same thing happens to chickens, I remember being gassed as a child for a operation and I felt shear terror all though it was only a second it seemed forever.