How did you come to this conclusion? This is so unbelievable wrong I don't even know where to begin.
Cats do not just kill weak animals. They kill strong and healthy animals at an alarming rate. Cats have been responsible for at least 33 island animal extinctions in recent/recorded times [0].
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?
That's an overly simplified model to the point of incorrectness. A Lion wants a sick / old / or young water buffalo because it's less dangerous. However, a house cat does not really care about a mouse or bird.
If you look at mice reproduction they mostly get eaten before adulthood. Young healthy animals are often preferred prey because they are easy targets.
Finally, predators regularly drive species to extinction, which is a large part of why the vast majority of species to ever exist are not currently around.
TLDR; Cat's need 8 adult mice a day if that's all they are eating and they really can't afford to just look for sick animals. They easily hunt healthy adult birds, some baby rabbits, or other small animals.
Where do you think tens of billions of old mice/rats that are to die yearly are supposed to go, especially in developed areas where there are not many other predators?
In predation even without any specific choice by the predator, the resulting prey is naturally skewed toward weak/sick/old, ie the ones who is slower, has lower attention/senses, etc. Until some age the young are naturally on the weakest side and not surprisingly may fell prey a lot. Such filtration results in more healthy/agile/stronger adult population.
Relative size makes a big difference. The most extreme version of this is krill, as a blue whale really does not care about the fitness of the millions it needs to eat.
What your missing is most mice don't reproduce. The average litter is 6-8 and mice can have multiple litters per year. So, mice basically never reach old age.
The selective pressure has more to do with caution because a completely heathy mouse can't out run a cat / owl etc.
Put another way cats rub at up to 48 Km/h mice 13 Km/h. Thus, 'fitness' has little to do with athletic prowess.
Cats do not just kill weak animals. They kill strong and healthy animals at an alarming rate. Cats have been responsible for at least 33 island animal extinctions in recent/recorded times [0].
[0] https://www.nature.com/articles/ncomms2380