And Doves. Also Eagles I suppose, and Ravens
They seem to crop up repeatedly in folklore, 'culture' and religion, particularly Cockerels and Peacocks I'd say

You're right, all sorts of birds are embedded in religion and folklore and have been for thousands of years. It's probably because birds can do something amazing, flying through the sky. Since most of the world's ancient Gods were thought to live on mountains, birds were also thought of as messengers between Heaven and Earth.
There's a story in the Book of History that dates back to about 1,200BC and is about the ancient Shang ruler Gao Zong that says something like:
"Gao Zong was a very pious ruler who worshipped Heaven daily with elaborate rituals but neglected his people because of this.
One day, Heaven caused a large male pheasant to alight in the courtyard of the palace where Gao Zong was worshipping Heaven. As the ritual proceeded, the pheasant mimicked Gao Zing's movements, strutting and posturing, screeching loudly whenever Gao Zong spoke.
After this, Gao Zong worshipped Heaven less frequently and returned to ruling his people instead".