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].
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?
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.
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.
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.
Modernity comes at a cost.
Do you have some sort of data to back this up?
This is likely unpopular (especially with cat people like me), but I'm not sure what's callous about it—unless you mean the "meh" part, in which case I'm not sure how backing it up with data makes it any less callous.
There are 10+ billion birds in the US and a smaller but very large number of bats and well over 1 billion of them die every year. Talking about hundreds of bird/bat deaths in that context is almost meaningless.
PS: A fungus on the other hand is decimating bat populations and represents a real threat. However the actual threat is not exactly making the news. https://en.wikipedia.org/wiki/White-nose_syndrome