Once again you show your lack of knowledge between the difference of pen chickens and wild roosters. Once you have learned this first hand then come talk to me.
One day I will show you pictures of a holed up pheasant and you will see how easy it is for a dog (especially those that hunt with high heads) to miss and tucked in bird. The only way you know it is there is because the grass or your boot makes a different sound as you pass over the bird.
There is a reason these birds last 3, 4, 5 seasons and its not because they run fast its because they know how to hide and they know where the escape routes are.