How high was the branch? They probably could have used a pole saw, and the DNR folks probably had one. They might not've had enough extensions; might have to borrow some from the local power company. Mine will reach about 40 feet, but it's getting pretty hard to manage that long.
Even easier would be to use a rope saw. Those don't really have a height limit, you just need a bow and arrow to get it up that high (please don't skewer the eagle)
Shot at a high angle, how much energy would be left in a 40 grain .22 bullet when it hit the ground, a half a mile or so away? They're not the most aerodynamic things. First thing I thought when I read the story was "what's his backstop?"
He used what he had, and the story had a happy ending. I suppose we shouldn't over-analyze it too much. (where did he get that much .22 ammo?)