Our neighbour few years ago took an air rifle to the local pigeons. Mrs MV had to phone her to tell her to stop after the fifth injured bird had hit our windows and left a bloody trail down the side of the house.
It's not big, macho or clever; it's brutal and ghastly.
Just don't fucking do it.
In fairness to the many thousands of responsible members of the UK airgun hunting community, you
can humanely kill woodpigeons with a non-FAC rifle.
The thing is, it's by no means easy and the only way you can be sure to immediately kill your quarry is with a head shot. The head is a softer target than the rest of the bird, but it means being confident of putting your pellet in a target area the size of a 2p coin at no more than 35m.
For this you need a good vantage point, a steady hand, a very accurate weapon and the willingness not to take the shot if there's any doubt as to whether or not you're going to make a clean kill. A lot of people who think they can bag a few woodys seem not to understand this, leading to the the sort of carnage that Mrs MV witnessed.
Which is, as you say, ghastly.