In short this is always a pain.
Small animals we butcher ourselves, bigger animals go to professional butcher. With the costs involved for processing these days, we will be moving towards self reliancy.
Either way, for me personally killing an animal is the hardest part. Not the blood, not the smells, but the act of taking life. Makes me feel down for the rest of the day. It is a part of the circle of life on the farm. A farmer sees things come to this world, takes care of them as best as they can, and in the end, they see their demise.
Now how many people think that they actually killed the animal every time they buy a piece of meat in the store? Not with your hands of course, but still an animal is dead so we could live.
And yet, 2-3 generations back it was quite normal for a Mama to call one of her kids to kill a chicken and bring it home to be cooked for lunch. What happened to us that we are so sheltered and dis-attached from all of this?
Interestingly enough the butchering part which can get really gross at least at the beginning with lots of fluids, blood, smells and all, this part is surprisingly easy.
Anesthetics are not being widely used. Aim is for quick and clean demise.