I have half a cat, a stray who won't enter the house even if we wanted him to.
I need him to control mice; I'm actually thinking of getting more cats. I'm in the case of the farming situation, though.
I'm concerned about the birds, but not immensely with this cat because he is mostly going after mice and voles, probably because it's easier (he is hunting for food, he's only coming to us for extra food, mostly when it's very cold). I actually never saw him with a bird, but it may happen anyway.
I'm a biologist, so I try to avoid emotional thinking and take only facts. I need that cat. I need to kill some wildlife, as difficult for me as it is, to be able to stay in business.
If I stop controlling pests, I'll be broke in one season, and my customers and myself will have to buy food somewhere else, where people likely poison mice (the most common here), killing the raptors in the process.
I'm not sure what I would do if I had a cat going mostly after birds, though. That would be a huge issue, as I could not keep them inside either (farmhouse.. always open somewhere...)