The thing about rats and mice is that if you have worked hard to keep them out of your yard/house, then it just means they are in someone else's yard/house (or really they are probably still nearby just that you don't see them yet). As far as I know, it is not "your" problem that the mice and rats are now at this other neighbor's place. He should make the same efforts to keep them out of his domain.
Chickens are great but some alternatives that might give you the same feelings of satisfaction are raising quail or maybe even jungle fowl, which are a type of chicken that are much quieter than a typical chicken. But I can't say if they're cleaner because, well, they're still chickens.
However, like someone said above, proving you can keep chickens isn't the same as keeping quail or even ducks, so even the quail alternative might not be valid if they aren't specified as being OK, assuming you might even consider that.
Being a chicken lover though, I'm disappointed in communities that allow dogs and roaming cats but not chickens. To me, roaming cats that go around in people's yards are pretty annoying, so I suppose someone might feel the same about chickens in your yard.