On the Outdoor Channel or whatever it is, there's a Brit show on acquiring and cooking wild game. A couple of weeks ago, the chef visited a farmer friend that was about to thin the flock of pigeons raiding his land. Shotguns were involved, along with a motorized pigeon decoy device.
That seems a lot easier and more rewarding than looking for bridges that need to come down that just happen to be the habitat of pigeons.
I imagine that an adult pigeon is pretty tough, but I’ve eaten squab which are quite delicious.