If you "rehome" an unwanted bird, the bird must suffer the stress of being removed from its family/flock, introduced to a new and probably hostile bunch - at least initially - and standards of care which may not be what YOU provide (or, who knows, could be better, but are you going to do a site visit first? Not usually, we're just happy to be rid of the "problem") and they endure a lot of stress just in the transport, etc. If it's a mean rooster, someone is going to either kill it or punish it for that behavior - and I've heard of some hideous "punishments". So I think it's so much kinder to quickly dispatch the bird, and I will say that the anticipation/apprehension is MUCH worse than the actual deed, especially if you have prepared a good sharp axe or whatever. It also puts you in touch with the life-death-kill-eat thing, which really should be part of our whole conceptual understanding of living on this earth. If you are going to have chickens, you have to come to terms with it eventually. It'll be ok, really. Good luck.
PS: this all sounds pretty lofty; it DOES help if the first time is after being viciously attacked by the "cullee", if he drew blood - even better! - because it strengthens the resolve immensely, and drives one to immediate and decisive action!