I don't know anything about narcissistic children, but I know something about the full blown disorder (NPD) in adults.
Adults that develop this disorder usually suffered trauma/abuse as children. This abuse is usually by a parent who is an NPD/sociopath themself. Often the abuse is severe neglect or abandonment. It can even be overvaluing the child. (This may seem counterintuitive, but it's the academic term for "spoiled rotten.")
The NPD seeks to destroy that which they did not experience. A form of revenge? Envy? I don't know why exactly, but narcissists are attracted to good people who are strong, honest, kind, generous, empathetic, loved and authentic. They didn't have any of that as children. NPDs are incapable of honestly expressing and living those qualities. Others who exemplify and embody those qualities are targeted. And they are then systematically destroyed.
Self esteem isn't the primary reason. Empathy is a significant factor.