Have you ever seen a sighthound take a rabbit? It's such a quick kill, the rabbit will know precious little about it. What do you think is a more humane, safe and effective method of controlling the rabbit population when it reaches the level that is decimating food crops?
I don't have sighthounds, I had Lakeland terriers. They were always under control when out, and on the lead if anywhere near sheep, poultry or wildfowl. They both passed the KC silver level "Good Citizen" test, which they wouldn't have done had I not had good control of them.
They were brilliant around children, people, horses, and gave cattle a wide berth. One of them took a couple of rabbits, in open countryside, but both times it was clear that the rabbit had myxomatosis and she probably did it a favour (myxie is a horrible, slow death).
I make no apology for the (considerable) number of rats she killed when we had a rat problem locally. She only ever caught rats in my garden and she probably saved me and my neighbours the expense of calling in a pest control firm. And being killed by a dog like a terrier, that breaks the neck with one shake, is far more humane than poison, which gives the rats a slow death from internal bleeding, or trapping, which can leave them maimed for hours until someone checks the traps and delivers the coup de grace.
The hedgehog was also killed in my garden, either last thing at night or first thing in the morning, when she was let out for a pee, and in the dark. I had no idea hedgehogs got into my garden, which was (and still is) securely fenced.
I am very much opposed to hunting with dogs for sport, but think they can be an excellent method of pest control.