First, what urbanites call "pigeons" are actually more accurately "rock dove", a species which has well adopted to feral existence and survival in urban areas (because there's plenty to eat which doesn't move fast enough to get away and more than enough shelter). While you wouldn't want to eat them, there's nothing wrong with their offspring (if exposed only to a less "gamy" diet).

Years ago, we used to bait/trap the adult birds in old vacant buildings to be used for a sporting event which PETA would certainly describe as beyond barbaric, "Pigeon Shoots", in which the birds are hurled by "trained" pigeon-chuckers aloft in front of shot gunners. While the serious casualty rate among pigeon-chuckers remains modest (small shot rarely decapitates), the live birds do provide more action than clays in skeet or trap setups.

Pigeon/Rock Dove raised at home (or the semi-wild "Barn-bred" birds) are succulent fare when eaten young ("Squab"), a real delicacy when carefully prepared. The adult birds are certainly edible, but in my experience work best in "Game Pies" or similar casserole/slow moist cooking environments.

Spitted squab, wrapped in smoky bacon and cooked over coals, are certainly the equal of mourning or white-wing dove (and once used to the ideal of "dark meat"), not far behind quail (or even partridge) in sauteed dishes.