Or you can just toss some food out and call it a day
This ideology that all stray animals need to be siphoned up and neutered on the spot is bizarre to me. Sure there is a balance to be struck as you can't have a hundred thousand cats in a small area, but this idea that stray cats and dogs are simply not allowed to exist and reproduce naturally is insane.
Have you ever lived in a place where there aren’t effective control efforts?
These programs aren’t about people feeling good about themselves. The middle ground you suggest IS neutering and spaying all the animals you can find. The more extreme end of control is culls. It isn’t something you can ignore without pretty serious second order effects.
It ends up being pretty grim for the animals and sometimes humans too when stray population isn’t controlled. I’m talking about the conditions these animals end up enduring, not the culling which is its own grim process.
As someone who had to help get a stray cat population under control the ugly way (limited budget in a developing country), I would much rather that strays get fixed before it gets there.
I live in New Zealand, where they are big efforts to cull non-native wildlife, including all mammalian predators. There are traplines through the woods here, and I remember seeing a Department of Conservation visitor centre with a diorama of stray cats being killed by a kill trap.
Locally we've got an overpopulation of goats, and apparently they're going to start shooting them from helicopters.
They also drop a poison bait called 1080 that kills mammals from helicopters in remote regions here.
Domestic cats and dogs are not native to the Americas. They are pets or work animals. The cats should be adopted if possible and made indoor cats or neutered to keep their invasive population down. Dogs too, you need to treat feral dogs as wild and excellent rabies spreaders.