Forgive the double post, but in the last three years, well:
You may have heard of Pigeon Simulator, a 3D game prototype that looks somewhat like Goat Sim: https://www.bossapresents.com/
There's also Grayland, a game where there's a war going on between humans and aliens and you're a pigeon that shits on things: https://store.steampowered.com/app/297190/Grayland/
...I want to stress that I didn't go looking for these games. They found me.
Also, although there's no poop, you have to admit Untitled Goose Game hits a lot of the checkmarks you laid out for bird mischief in the English countryside. Put all these games into a cauldron, stir them up and you might have the perfect birdshit game on your hands. Please wash your hands.