Why do so many Adoption Centers refuse to adopt smaller breed dogs to homes with children?

My family is currently looking for a shih tzu. I adore this breed and have been researching them for about 3 years. We are going to be completely ready to have one in March. I would love to adopt one from a shelter, but most will not even look at my application because I have 2 children under the age of 12( 10 and 3). I feel that I am going to be forced to go to a breeder for a puppy. I understand that many families do not supervise their children around their pets, which leads to injury to either the pet or the child. What I don't understand is, if we have excellent vet references, a large fenced yard, and are willing to undergo many home visits from the shelter, why won't they consider our application? Why should all families be judged by a few bad examples? I think it is extremely unfair that I will have to buy a puppy from a breeder rather than adopt one into a loving home where they will be loved and doted on.