The thing that bothers me about the throat-slice method is the fact that the bird doesn't die instantly. In that case, they are likely flapping because they are bleeding to death, not because they are already dead. The thought disturbs me quite a bit.
I could never kill any of my birds...I know in some cases it's the only option for some people and I respect that, but I could never do it myself. I don't even think I could send birds off to be slaughtered. Maybe that makes me a weenie in that sense, but I can't help it. lol
I'm really sorry you had such an awful experience. That would bother me for quite a long time, too. The rooster is in heaven feeling fine, and I hope you start to feel better again soon.