Well, we've got one of each ....
... and a parakeet to boot; it keeps the kids happy.
But when I walk the dog, I pick up after her. The cat is a house cat. In fact, her lack of front claws (yes, I know, how cruel!) kind of requires that we keep her in for her own safety.
I would no more let her crap in the neighbor's garden then I'd let my dog allow one in to crap in ours. As to why anyone would keep a cat (who showed up before the other two) I can only speak for us but I'll give two reasons: 1) She needed a home (feral kitten), and 2) We have a fair number of mice in the yard. AFAIK, the difference between a field mouse and a house mouse is strictly a matter of residence. Oh and field mouse: cute; house mouse: not so cute. The mice haven't been clued in about the cat's lack of claws of course, her litter box smells the same.