I live in the country, so I use the .22 method, just processed a BR yesterday. This is not a method for everybody especially if you live in the city, a no no. You must be trained in safe firearms use, in my case a former LEO and firearms instructor, and as a youth dispatching cows with a .22 in a slaughterhouse. But then I was well trained with a firearm at a early age, NRA youth safety and marksmanship, and a JR CMP. When it was still allowed as part of a school curriculum(private school).
The death is instant, if quick enough the feet can be grabbed as the death throws are delayed, by a second or two after the shot. In this method the chicken never suffers, not even to be chased, I always do the shot while the chicken is foraging from the back of the head. It takes patience to wait for the right shot just as in hunting. I then tie the feet and hang the chicken to drain the blood with slicing the throat(just as done in the slaughter house).
This method is ONLY for the very few that can place a shot into a small area and are very aware of the dangers and safe use of a firearm. ONLY to be done in area that is not populated. Shot placed from the rear at a downward angle into the ground as the bullet will exit the skull. Use .22 short if possible less noise and it is enough to do the job. This is not a good method for processing a lot of birds, it is too time exhaustive in waiting for the clean shot.
In the case of a large processing I would think using a cone and penning the birds in a crate while they sleep that morning would be better. Or breaking the neck. Chopping heads seems to me like a good way for an accident.
I have known people with HIGH powered pellet rifles to take coons, rabbits, and possums with them and they drop like rocks. BUT these rifles have velocities of 1200fps NOT a cheap pellet rifle. Mine is strictly a target pellet gun and reaches no where near enough velocity to efficiently kill a animal outside maybe a mouse. I may in the future buy a high powered pellet rifle as it would be cheaper and quieter.