If you're a little squeamish (and we all are when we first learn to process chickens) the hatchet sounds like a way to get the deed done quickly. Once you get through your first 5-10 chickens, you'll feel much better about the whole process. I originally got chickens/pigeons and became a falconer. because I love animals, especially birds and appreciate life. What a harsh reality to discover that there is so much killing/death involved with keeping any quantity of animals. Prior to this, I've only had dogs, and the occasional small animal. In small quantities, we can fool ourselves by really isolating the death part of the equation.
Coming to terms was really a good thing for me. It wasn't as bad as I had thought. Once you get to the point where you've processed with a hatchet, I'd recommend trying to use a killing cone and bleeding the bird out. There are some added advantages of having the bird's heart pump all the blood out of the body. First it tastes better, and second the meat keeps longer. So if you are ever planning on keeping meat birds and storing a bunch in the freezer, you'll get more mileage with bleeding them out. You just need their heart pumping for as long as possible to get out the most blood.
I know it sounds bad and I didn't think I could do it either. I thought it would be a slow and awful death but it really isn't. The trick is to use a really sharp blade. I got a scalpel with disposable blades. I've also used the kitchen knife to do this, but you really need lots of pressure unless you have a razor sharp edge. That could add to the suffering for both you and the bird. The first few chickens you bleed out, pluck a few feathers from the area you're going to cut. After you've got the feel for it, you won't need to pluck any. There is a sort of calming or relaxing effect that holding a chicken upside down or on his/her back has. My wife refers to this as "Tonic Immobility." They really don't know what hits them. You put them in the cone and they hang upside down and you just wait for them to calm a bit and after a few seconds, they're at peace with the cone. Then you make the cut into their neck and if the blade is sharp, they don't really seem to know they're being cut. Then the blood comes out of their body and they drift off. I think it's probably much more disturbing for the chicken to be held down and have it's head cut off. Chicken don't like to be restrained by hands, yet the cone doesn't seem to bother them as much. Also I'm sure the chicken is aware, at least for 10-20 seconds that it's head isn't attached. The blinking eyes of a beheaded chicken kinda freaks me out. I'm of the mindset that if the bird dies by drifting off, with it's body relatively intact, they'll be more at peace with the ordeal. So while the "DEED" may be over more quickly with the hatchet, I think it's more traumatic to the chicken's senses because you've got a conscious bird that you're not only cutting, skin, you're breaking bone and severing the head. I save the dismemberment for after the chicken is dead.
So those are just my thoughts. I'm not advocating that you use any particular method. I'm just describing the way I do it, and why I do it that way. You'll find your own way that is right for you.
Dan