Most people with schizophrenia are not violent and they prefer to be alone and are very withdrawn. There are 3 major types of schizophrenia as well and most have very different behaviors. In fact, if this was rare psychotic break or act of violence, there is no way the son would have any type of organized thought process going. That means, the crime would have happened in the home...it would be obvious even to untrained people and the son would not have had the capacity to hide a body. He would be too confused and disoriented to have a "perfect crime."
I would be more curious to know about the people who lived near or around this family or in the town. It would be an easy crime as most people and media would focus on the dysfunctional family as suspects and not on the average person who lives down the block. Unfortunately, this little girl with self-esteem issues and an unstable home life would be a very easy, compliant target by someone else. The family would be the perfect cover-up, just being who they are and with their daily struggles.