First thing's first. It's not just about the poop, although the potential stank of hundreds of cats living together in a single dormitory is a haunting prospect. As was widely reported at the time, J.K. Rowling has already addressed the wizarding community's approach to feculence. No, the problem here isn't so much nature's call as it is what Steve Irwin referred to as “nature's way.”
As most people are aware, cats are what's referred to as “obligate carnivores.” They gotta eat meat or they'll die. The same goes for toads of all stripes — they're carnivorous, and not particularly picky about which creatures they consume. Owls, meanwhile, just go ham on any animal smaller than an owl, swallowing their still-twitching carcasses whole and then puking up the bones later. The Hogwarts pet policy has the potential to turn any given common room into a zoological Hunger Games. Ever been in a public space where someone's emotional support pit bull caught a glimpse of somebody else's therapeutic cockatoo? Imagine that, and add dozens of panicked children with magic powers trying to save their pets. It's a nightmare, and that's before you factor in the way that some of the teachers just turn into animals sometimes. How many points does Gryffindor lose if your bird tries to eat McGonagall?
Then there's the public health issue. Hagrid mentions in the first Harry Potter book that cats make him sneeze. Translation: Allergies are still very much a factor in the Wizarding World. The school grounds are a perfect storm for youth anaphylaxis, and given the fact that the infirmary looks just advanced enough that it could be used as a quiet place for injured World War I soldiers to slip into death, the chances of a magical Epi-Pen being available seem razor thin. Say what you want about Voldemort, at least his snake was hypoallergenic.
Chalk up one more reason that it would suck to live in the Harry Potter universe: The best school in England is a Thunderdome pet shop full of wheezing children.