I was careful to say "simply for the fun of it". I am not a vegetarian.
I wouldn't shoot an animal for the fun of it, so I agree with what you are saying.
Unfortunately, the HSUS spends tons of money trying to outlaw the legal hunting of these, and all other species. So they are in effect, telling others they can't shoot and eat doves.
Hunters don't kill just for the fun of it. They hunt for the fun of it, and the fun of it is much broader than killing.