Yes the bird is hanging upside down facing away, restrained in a similar way as a killing cone, but I used an old kitty litter jug I had, and built (well my husband built and I tweeked) a simple 2x4 frame to hold it steady and at the right angle. Yes a revolver type would be easier to get the angle you need, however to get the velocity up at a price I could afford I had to do the rifle. There is no need to cut the throat w/ the shooting in the head method they bleed completely out through the wound and just as fast as a neck cut.
Oh and you are absolutely right about taking your time letting the bird settle into thier trance thing so they are relaxed and not moving, make your aim true you have plenty of time, so don't rush it.