In any discussion of cats that study is mentioned and usually mentioned several times at that, and that is not surprising giving that it is published in Nature and comes from Smithsonian - it immediately generates a lot of trust in it.
2 things are notable about that study:
1. the researchers didn't specify the proportion of ill and old birds and mammals in those billions of killed. It is a glaring deficiency for such a scientific study. Billions of mammals and birds reach end of life each year. In natural settings natural predators take care about that. In developed areas basically only cats and rats are such predators who is available to perform that duty. When it comes to the ill birds and mammals - we have already been through such situation with deers and wolves. Thus, if the proportion of ill and old is high then it would mean completely opposite conclusion than the one the study reaches - it would mean that cats predation is beneficial for those birds and mammals.
Now, why the study has such a glaring deficiency? One can make an informed guess when one learns the 2nd notable thing:
2. a researcher in that department that produced the study was convicted of animal cruelty committed toward cats and that researcher is cited in that study.
Some other things to note wrt. cats and birds :
1. rats, a human civilization companion, raid birds nests and eat eggs, i.e. causing massive damage to population during the most fragile stage in the birds lifecycle
2. cats don't eat eggs
3. cats control rats population
Anekdotally - my grandmother's farm (farmhouse and 2 large barns) had always had a bustling population of barnswallows, and it also had always had a pack of feral cats living there. Sometimes we'd find a small pile of feathers - all what is left of an ill or stupid barnswallow as the cats would have no chance of getting a healthy flying bird. The farm had naturally had a lot of various food stuff - cattle feed, pig feed, chicken feed as well as human food - pretty much everywhere, ie. in storages as well as being actually fed in the barns and outside ranges to the cows, pigs, chicken. Despite all that food stuff present in large amount in many places, there had never been any traces of any mice or rats (note: no artificial pest control measures - like traps or poison - had ever been taken on the farm). So to me the cats are necessary companion of our civilization, and as far as i see they are beneficial - by protecting from rats and by taking out ill birds - to whatever birds population still surviving the stress of development already imposed on it by the humans.
I'm not here to tell you I know the absolute truth on the matter, but I can say definitively that your assertion of this fact extends beyond conclusive data. Cats decrease rat sightings, but urban rat populations are good at evading them [1, 2]. Where cats do prey on rats, it's usually juveniles with ultimately little effect on local populations [3, 4].
It's easy to imagine that a certain percentage of the birds cats are killing are ill or old. Yet in the absence of actual data, your assumption that this is the bulk of bird deaths seems like obvious wishful thinking. It's also just as easy to imagine many of the deaths are juvenile / nesting birds too. And unlike rats, most birds can't repopulate so quickly.
"We estimate that free-ranging domestic cats kill 1.3-4.0 billion birds and 6.3-22.3 billion mammals annually."
https://www.nature.com/articles/ncomms2380
Honestly, birds are pretty much the only surviving wildlife in many areas and I wonder what our children and grandchildren will encounter.