Am I missing out on the spirit of this thread if I actually answer the question? OK, I won't answer the question, I'll answer the other questions.

I've heard one well-expressed, well-reasoned argument as to why killing foxes by hunting with dogs is more humane than shooting them, and one badly expressed but equally well-reasoned. If you want to shoot a fox you need to use a shot gun to be sure of hitting it; using a rifle is going to mean a lot of wasted bullets and you probably won't see the fox again after the first bang. (Also, shotguns are readily available in the UK; rifles aren't without a lot of hassle.) The fox will almost certainly not be killed immediately, but will disappear from sight and die slowly and in much pain, possibly over several days, and maybe from starvation rather than its wound.

On the other hand, a pack of dogs is very good at catching the fox in the first place; and the fox will die there and then, in a short time: no more than a few minutes at most. It will be unpleasant and painful--no argument there, but if it has to die at all, this would seem by far the lesser of two evils.

Do foxes need to be killed at all? I've no idea. I know on occasions they have to be conserved to keep the hunt in business. And we should distinguish between the idea of a lot of people whose idea of fun is to tear about the countryside on horseback at high speed (which sounds pretty exciting, actually) and the idea that the fox's death is part of the fun.

And foxes aren't Nature's mass-murderers. Their instinctive behaviour as predators is to catch something and kill it; all the other lambs/geese/chickens will run away and be safe; the fox will take its prey home and eat it. But if the prey animals are all penned up and can't escape, the fox keeps on killing till they are all dead, because he can't stop until they are.

Any questions?

Steve