We should be exterminating feral and "outdoor" cats.
I need to preface this by saying that I do not hate cats. I have never owned a cat, but I have family members who do, and for the most part they have been loving and wonderful animals (except for my cousin's first cat Cinderella, who was an asshole. Literally every other cat she's had has been lovely)
But I live in an area of the world where, just like every other, feral and "outdoor" cats are a major issue. I see them prowling around every day -- I hear them fighting at night, and I see the consequences of their introduction in the corpses they leave behind.
Domestic cats kill, by the estimates of recent study, literal billions of native mammals and birds yearly in the United States alone. This is not even considering their impacts on reptile, amphibian, or invertebrate populations, which are generally poorly studied and understood. They are the number one human-caused driver of biodiversity loss we are aware of.
And yet, we do nothing about it. We let them roam free and any attempt to stop them is seen as fundamentally inhumane and immoral. Any attempt to get cat owners to keep their animals inside, like literally every other pet, is rejected as unnatural. It annoys the fuck out of me.
These cats are unadoptable and even if there were, there are literally not enough households which would take them in. Catch-and-release sterilization has been tried and it is entirely uneffective and cost-prohibitive. I do not like the idea of killing any animal, big or small, but when given the choice between our native animals and the cats we introduced through carelessness and negligence, I choose them.
So we should be exterminating feral and "outdoor" cats. We should fix the problem we created.