look up Australian Cat Plague, if you want to see what happens when you truly let cats just go ham outside "because it's natural" or whatever reason most people use. they're little apex predators that kill anything they can get their claws on, if there's no native predators to control them they will dominate everything smaller than them

you cannot save them all, no amount of adoption centers would be able to receive the inflow of cats from a proper plague of them. there entire pet industries designed towards selling these animals to people who probably can't even take care of actual human kids let alone an animal