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One big difference is that dogs are predators that we have encouraged to propagate for our own uses and desires. They are human caused and it's on us to protect our birds from OUR predators (the dead chicken around the neck is usually reserved when a person's own dog kills the chicken).
Also, dogs and birds think completely differently. They have different motivations. Dogs, as a domestic animal, generally very much want to please. They also have a concept of "rules" from their lives among humans. Also, they are usually not hunting out of hunger but out of instinct or for fun. Eagles hunt for food, period. There is no such thing as a rule for eagles. If it moves, it's fair game. They also generally hunt from the sky, and probably don't distinguish a chicken from any other type of bird, nor can they. So no, there is no way that is an effective method of predator control.
For the trouble they went to to set up the situation of the eagle being attacked, they could have put up a stupid fence made out of sticks and netting. It wouldn't take much. I think they were just mad and being evil and unreasonable, looking for revenge on some poor bird that was just being a bird.
Edited to add that they are going to introduce a disease vector into their flocks from the raptors and kill off more than the predators would. Dumb idea.