One thing a lot people don't understand is that clean does not equal free of infectious agents and that dirty doesn't not equal to infectious agents. Clean does not mean safe, dirty does not mean unsafe.
Pets, in general, are FAR safer to people than other human being in regards to infections (and physical injuries too). A dirty dog pretty much cannot give ANY infections even if he lick you in the mouth. Shaking the hands of some splotless guy in a suit can give you serious infections (flu, gastro-enteritis viruses, etc).
We just don't share many common pathogens. Their bacteria and viruses and mostly harmless to people (and vice-versa).
Of course they are exceptions, but they are not common. Ask yourself how many times in your life did you get sick because of something a pet gave you. How many times did you get the flu, stomach flu, cold, STD, etc? I think I can safely say that the second is by far more prevalent.
I work with animals (I'm a vet), when I poke myself with a dirty (contaminated) syringe from one of my patient, I have ZERO worries I'll get sick. I am far more likely to get sick from the owners of my patients and my staff than my patients.
We deal with and learn about zoonotic diseases (disease than can be transmitted from animal to human). It's one of a vet's job to minimize zoonotic infections. There is really not that many that we have to deal with.
Dogs do smell, so does people. People that are not used to the smell of dogs will notice it. People that live with dogs often won't smell it, they become desentitized to it. Some people that are reclusive will find that people stink.
While fur might be messy, there is zero harm in it.
There is mounting scientific evidence that the rise of allergies and immune disorders in people is cause by the lack of exposure to various organims. Living in a aseptic world actually make you sicker. Most organims are not harmful, and exposing your body to them make your immune system more 'balanced'.
There are studies that shows that having kids growing up in a multi pet household decrease the chance of them becoming allergic in general.
Germaphobes and petphobics are actually harming themselves instead of protecting themselves.