So if you had a dog or a cat that was seriously ill and had to be put down, I wonder what you would do, Chickenfan? Much larger carcass, much more difficult to kill painlessly and without distress, except by injection by the vet.
I would also bear in mind that corvids are among the most highly intelligent of birds, much more so than chickens, and I would be as reluctant to kill them as I would be to shoot my own hens. It would spoil my view of the peaceful garden if I hung up dead rook carcasses as well. I would just keep my growers in a safe pen, for protection against other predators as well as rooks.
Please understand I'm not getting at you, Chickenfan, I understand you have a genuine dilemma here, and you've brought up some important questions about our relationship with the ecosystem that we all should consider. Personally, I've decided that if any of my hens becomes sick with anything serious enough to need an antibiotic, she will either be dispatched then and there, or possibly I will try simpler treatment methods for a while if she doesn't appear to be in pain. There's a very real risk to human health from the over-use of antibiotics in poultry.