In most areas of the US and Canada, it is illegal to hunt birds with a rifle or pistol. It's completely illegal to hunt migratory birds with a rifle or pistol in the US and in all but the NW Territority of Canada. It's primarily due to overpenetration issues with a bullet or slug through a bird vs. neighboring humans.
Since most birds must be airborne in order to legally shoot, even Annie Oakley would have some difficulty controlling where her bullets were actually coming to rest, as many bullets travel over 2300 meters before landing (over 5000 for a 30-06 rifle), where a shotgun's effective range ends at 45 meters.
BTW, I am not a hunter, I just grew up around several people who are. Hunting for sport never really piqued my interest, although I did try it. I do however enjoy target shooting. You should try my recipe for roasted paper target with veal demi-glace. It's a little dry, but full of fiber.
Best Regards,
IG