We have a pigeon on our balcony. We picked him when he was only a week old and now he lives with us. Our balcony is isolated from the street, so every morning we open a window there so Theodor (his name; although we're not sure, maybe he is she), so Theodor can go out and fly. In the evening he returns; sometimes he gets stuck somewhere due to rain (they do not fly in the rain) until dark (they do not fly in the dark either) and returns in the morning. We worry a little when this happens.
He is not strictly a pet, we try not to turn him into one and hope he will find a spouse in the spring. In the meantime we'll build him a small house that may serve as a nest if he is so inclined. We've read a few books about them and joined a chat of people who keep pigeons, en masse or just adopt a pigeon who needs it.
We do not do that often but about once a week we catch him and sprinkle with a dust against parasites. Pigeons are very pleasant to touch. The feathers have a silky feeling and they are warm, warmer than people.
There are a lot of excrements, of course. We covered the balcony with cardboard to protect and wear a dedicated pair of slippers. Yet we see these little piles with satisfaction because they mean that Theodor eats well and is healthy. By the way it must be a good fertilizer; old books say it was a source of income for the pigeon keeper.
So maybe you can make friends with pigeons instead of shooting them. The balcony is too small anyway (see "The pattern language"). Give them a bath (they love to bath), make them a feeder. (You can strategically position it so that excrements will mostly fall outside.) They are very lovely creatures.