The thing about cockroaches isn't just that they're incredibly gross and/or disease-ridden, but that they insist on living right in our kitchen. Most people would rather see a freaking ticking time bomb in their cabinet than a nest of these *******s.
But the reason why roaches survive so well around your boxes of twinkies is that for centuries they were like our tiny, industrious little roommates. Cockroaches specifically adapted to share the nests of larger mammals, getting access to a continuous stream of food scraps, mold, mildew and even the delicious eggs of more dangerous insects such as fleas, bedbugs or lice. In return, us larger vertebrates are supposed to enjoy the free janitorial service and snack on the ones that don't run fast enough.
For most of the animal kingdom, it's still a pretty sweet deal. But a few centuries back, we humans decided we could keep ourselves clean without an army of hungry bugs, and our former custodians became just another form of "filth" to our high and mighty standards.
And just how filthy are the little freeloaders? Well, that depends.
If you happen to live directly over an open sewer or keep decomposing corpses under your floorboards, it's entirely possible that your cockroaches might be tracking the occasional pathogen on their sticky little feet, but their habit of incessant, cat-like grooming tends to rid them of contaminants long before they might scuttle over your ham sandwich.
In fact, testing shows that germs don't even stick to them that easily in the first place, so they're really only as dirty as whatever they're standing on at the time.
So, we know that roaches just aren't very dangerous... but we would still be better off without them, right? Sure, if you don't mind the wafting diseases that'll build up when we lose a major player in the process of decomposition. Cockroaches are also the primary predators of bollworm and armyworm, two of the most destructive pests of cotton, soybean, corn, cabbage and tomato crops in the United States and Africa.
Really, it's just a PR problem at this point. Maybe Pixar just needs to make a movie starring one of these guys.