cddt:
Tinkerisk:
I'm sure there are a few strictly protected animal species in your area that are not possums and can get trapped.
Morepork, fantail, keruru, tui, kaka, etc.
But if you don't trap the pests (in urban areas primarily rats and possums), there won't be any birds, because they simply won't survive long enough to leave the nest.
Trapping roaming cats would be helpful for native species too, but is socially frowned upon.
Our cat has brought in zero birds in the last 12 months but maybe half a dozen rats, so is a net benefit to birds.