The use of nitrous for cavities, especially with children, is common and perfectly safe. It would take a very extreme case of malpractice for a patient to die from it. As young as your child is, it may be worthwhile to take her to a pedodontist, but there's nothing wrong with taking her to a general family practice dentist. A pedodontist will almost definitely give her nitrous. But, like I said, it's perfectly safe and will probably make the experience much less traumatic for your daughter as she won't really feel any pain and will be less aware of the strange things going on around her.
The reasoning behind the dentist not wanting parents in the operatory with the child is probably because of the common tendency for parents to get upset, which further upsets their child, making the dentist's job more difficult, or in some cases, next to impossible. That being said, I don't think doing so is good as a general policy and a pedodontist would probably not do it.