We used to take our family pet to the City all the time as well. No one seemed to mind and all the pets that I remember being in the city were well behaved.
Parks that serve food and have high traffic areas restrict animals to the service animal category as HB said. This is a theme park standard. You won't see the City reversing this anytime soon....if EVER.
Mind you, there are at least nine types of service animals. there are seizure dogs, comfort animals for those who have agoraphobia, dogs for the blind, and yes...there are seeing eye monkeys. These are all protected under the Americans with Disabilities Act. Not much you can say or do about it.
Mind you, there is also something called an MR (Manufacturers Requirement) that all attractions have (Ride Law Legislation is my field at Disney). Most of the time there are restrictions on service animals permitted or not on the rides. Wildfire, for instance, is one of them I'm sure. You will never see a dog on that ride or many others.