In a way it's strange that two of the three biggest pest bird issues came to the new world from the UK where they are now rare.
Consider the gray squirrel you openly hunt as an invasive species taking over from the reds. They're game animals here, seasons, limits, hours, baiting prohibitions the lot. Same as say deer (although you don't need tags like for deer). Likewise Rabbits and Hares are game animals. Of them only the Jack Rabbit has open season and no bag limit.
Magpies are prohibited, canada geese (which migrate through here) are seriously regulated and not legal for airguns. As are crows (sport hunting with airguns is prohibited in all states). We don't have rooks here.
Hunting is a much different thing here, the rules are much different reflecting the needs of Game Management. Traditional hunting is not reserved for the rich, millions of common citizens take tens of millions of deer (for example) each year on public lands. Even more waterfoul hunters and ducks and geese taken. Airgun hunters are rare indeed. Under a percent if I was to guess.
FWIW, blackbirds are generally protected as well except in rare cases like this farm where there is a Depredation Permit in force covering them. They can show up a thousand or more at time, the owners see no immediate shortage. In fact they invite us down to shoot a few while we're also after the ground squirrels (number one pest, cost wise, in the state.....more expensive to crops than insects).