I was pressed for time on my other answer, I'll give a little more input here....
My Honey is also a hunter, and grew up roaming the local vineyard with a pellet gun to control the rat population. He's a very good shot
He did dispatch our birds for a few years with a pellet gun. We're not allowed to shoot "real" guns where we live, so pellet it was. This wasn't large batches of meat birds, this was a few overflow cockerels now and then.
As I said, it's a small, unpredictable target. If you can drop the first one with no fuss, the rest don't get too riled up. But if the first one does the flop, you're going to have a ruckus in the pen and have to wait until they settle to get another good shot.
Getting a bad shot...well, we all know that's not a great thing. You're going to have a knife anyway for butchering, so it will be available if you need to slit a throat or decapitate after a bad shot.
I had no problems with this method, except I had to rely on my Honey being available to help me butcher. I decided last spring to learn to do it myself. He's a
great shot---I'm okay but knew I could never be this accurate. So, I use a milk jug and a sharp knife.
Like so many other things with birds, I say try it and see how it works for you. It may be the best way ever, it may be not so much. Honestly, the bird's going to wind up dead either way, and you'll learn what you want to do next time.