The American Pit Bull Terrier has been bred for generations to be good with people. In fact, I compare their attitude towards people as being pretyy "lab-like", they should be indiscriminately friendly. Now, they may not feel the same towards other animals. They are, like many of the terrier breeds, more prone to dog aggression. Not every APBT is, but anyone considering this breed should be prepared that their pet, even if well socialized, may mature to be intolerant of other dogs. On the other hand, being aggressive towards people is way outside the norm, and considered unacceptable.
APBT's are no more prone to "snapping" than other breeds. True cases of dogs "snapping" are very rare. More common is the case of dogs that were giving off plenty of signals, but their owners were not reading or respecting them. If you had your schih tzu's out and around, you've probably run into people who claim (and believe) their dog is friendly, but you could tell by its facial expression and body language that it was not. So no, APBT's are not prone to "snapping". No dogs are.
I would recommend that you might try meeting some bullies in person. Is there a pit friendly rescue or shelter in your area, that you and your wife might visit? I commend you for doing your research. We got our first APBT when my son was 2 1/2, because we wanted a breed that was good w/children. 20 years later, we've still got them.