I do find birds, mice, frogs, and bats in the house is a nuisance, how the cats get them through the cat flap I don't know. Feeding birds varies town to town, I know in Chester it is illegal to feed food to the birds except for one small area in one car park, many seaside towns also have a ban on feeding food to birds.
As to feeding birds as food to other animals not so sure, the problem is some birds are protected, but not able to teach my cats not to hunt them.
At least in this country the birds don't seem to hunt cats, I remember watching the Caracara on the Falklands playing cat and mouse where the cat was being treated like the mouse. I suppose the red tailed kites around here could attack a cat, but not seen it, know when they are about as the crows make a racket, and try to drive them away.