Isn't that how the buffalo almost made it to extinction?

And the dodo bird did?

Pigeons don't harm anyone.  They have a right to exist as well.  Or did I somehow step into 1808 instead of 2008?

I know very well why cats "have fun killing".  It's because their neurons and their optical system are adapted to react to movement so they are hyperalert to that.  You might think it's enjoyment of killing, but it's just reflexes that allow them to thrive as predators.  That doesn't mean one has to let them kill things.  They can get just as much exercise in trying to run down birds who know they are coming because the cat has a bell on its collar.  In fact, they get a lot more exercise this way and no birds get killed.

Millions of birds are killed unnecessarily every year because of domestic cats.  And, because of other problems such as insecticides, pollution, etc, the number of birds is dwindling already...