There are 10B birds born per year in the US. In Nature pretty much everything is a food for the next level in the food chain, and there is no family&friends funerals. The more weak/ill/old - the higher probability to become predator's dinner. We don't have dead old birds littering the streets of cities. Somebody takes care about it.
And as you see from your own link cats mostly eat smalls mammals. Around humans it is mice and rats mostly. Rats do enjoy bird eggs and there is no protection here, until of course somebody takes care about the rats, like for example cats do.
As result cats do positive service to the birds population. The population which somehow manages to survive the urban and agricultural development - the main dangers for the wildlife causing population decimation and extinctions.
And island extinctions, really? So the cats brought by humans should have just committed suicide by starving themselves? Of course things can easily go wrong in any small system when the system gets severe kick from outside. And humans do just that to various eco-systems. How many species on various islands got extinct once humans reached those islands?