I would assume the bird knows not to damage itself I mean in the wild you see raptors carrying away serious animals.
Plus the bird is more autonomous and pics targets off where as a drone would to have to be piloted
The birds are pretty unfamiliar with spinning rotors, I don't think there is anything within their nature that would prevent injury. The autonomous part is definitely an advantage, plus an eagle or hawk could easily out-pace even the most substantial drone. This would be the big issue with the net idea.
At the same, even if it's a deterrent that would be enough. I'm sure most people who fly inappropriately aren't meaning to do so, and seeing a huge net attached to a big drone coming after your piddly toy should be a pretty good clue you're breaking the law.