It may seem silly to say it, but animals are not people. Animals didn't have any concept of sexual identity or ego. Neutering a male dog or cat will not change its basic personality. It doesn't suffer any kind of emotional reaction or identity crisis when neutered. A companion animal's personality is formed more by genetics and environment than by its sex hormones.
Medical evidence indicates just the opposite.
In fact, the ideal time to spay your female cat or dog is before her first heat. Spaying at a young age prevents uterine infections and reduces the incidence of mammary cancer. A spayed female is a more relaxed companion. What you will teach your children is that animals can be created and disposed of at our convenience. If you want your children to see the birth, they should also know about the death of the thousands of animals brought to our animal shelter.