Professional pigeon pest control is done with a hide a lure and a shot gun from the hide, which may only have to be hedgerows. The farmer provides as many cartridges as you want because the pidgeons eat their crops. But even getting a shot gun license is harder today, I know a farm where one of the sons can’t get a shot gun license because too many in the family already have licenses.
Shooting pigeons with air rifle is one pigeon at a time, the rest fly away and dont come back soon enough for you to get another. Its not viable professionally.
In a different world, Sanctuary Housing when I worked for them all council estates, have a few guys with an air rifle and they shoot out the window down the back garden. But its legally sketchy. But people living in that way get away with whatever they want.
There were two brothers who were good poachers they had an illegal rim fire 22 consealed in the bottom of a trolley and a three leged Lurcher which could take down a deer. They went up the beck at scalby not far from their council estate. Around the beck were game birds and deer. All they were doing it for was to be able to eat. The trolley allows them to get the deer home.