For one reason, they are terrible for the ecology. Wind turbines have been getting a lot of hate due to them killing 234k birds per year in the US. However cats kill 2.4b birds per year in the US.

https://cdn.statcdn.com/Infographic/images/normal/15195.jpeg

I'd rather have an overpopulation of cats than rats. Looking at you, New York, London and Paris.

You answered your own question. A lot of people spay and neuter.

In US, any cat that touches any shelter get spayed and neutered.

Some shelters are also kill shelter. If the cat doesn't get adopted, it will be put down.

Sometimes I wonder if we would accidentally make cats extinct.

Cats breed fast. Sexual maturity is 6-9 months, pregnancy takes 2-3 months, and a mother cat can go into heat a month after giving birth. Litters are mostly 2-6 kittens, and there is no such thing as "cat menopause" - your ancient housecat well past the end of a wild cat's lifespan can still get pregnant.

As long as there's a few breeding pairs out there, they can repopulate quickly. You would have to go to a very deliberate effort to make them extinct. Or accidentally destroy everything they would eat - which, given that they will eat pretty much any animal they can take down, would be a serious global extinction event.

House cats are insanely adaptable, it would be incredibly difficult to wipe them out. There are feral cats on every continent and while they are incredibly well adapted to cities and people, they also do just fine in the wild.

I thought it was clear with my second sentence. My question was facetious and rhetorical.