Lots of feral cats are shot on site too, mostly in farming areas. We have lots of problem introduced wildlife here, so do many places I suppose.
Unfortunately there's not a great deal that can be done in the arid interior. Feral cat territories can be very large, so finding and dealing with them is tricky. We actually have a number of sanctuaries owned by a private company that have cat and fox proof fencing surrounding the entire area. And I'm not talking small sanctuaries either.
On another note, I'm very keen to see a wild dingo one day.
In studies done of cats, it shows they have a 2 mile territory radius!
As far as feral and socialized cats being a threat to the bird population, they are way down the list of being the result of declining bird population. In fact TNR people are a solution to the declining population because most TNR groups take in socialized abandoned cat they trap and vet and adopt them back out to indoor only homes!
Here is a good article on what the major threats to the bird population are.