You can't get rid of the wild bird problem unless you get rid of cat ownership entirely.
Hmm... The decimation of the sparrow population is seen to be the result of the numbers of domestic cats, particularly in cities, not the existence of domestic cats per se.
So, actually, if people better understood the true nature of these 'passive' 
creatures, then maybe the bird population would be a little better off.
Dogs can be prevented from killing. Cats can't.
Dogs should be prevented from killing domestic animals, obviously, and where it's not possible to control dogs' prey-drive, they need to be kept on leads where there's a danger of them attacking domestic or protected animals.
It's not possible to train all dogs to leave smaller creatures alone. As everyone knows, there are huge numbers of terriers in rescue centres etc. Many of these dogs haven't been socialised around other animals, but are still great with people and (sometimes) other dogs.
Yes, you could argue that such dogs should always be muzzled/on lead, even on their owners' property, (or that they should be deemed 'dangerous', destroyed, etc..) so that the cat/rodent population can wander freely on any private land they choose.... And if the owners of those dogs happen to love the birds in their gardens, and those birds get killed by local wandering cats, then what the hell? because cats can't be controlled, right?
BTW, there was a great thread started yesterday about working animals. It's a great reminder of how useful and intelligent animals can be, as can be their relationship with humans, when people accept that there's a great deal more to them than their being 'passive' fluffy moving-cushion-things to decorate the suburbs and demonise from time to time when they show signs of being what they were bred to be.
Anyway, sorry for derailing. 