Ultimately, the family made the right decision for them. A media circus at this time would be too much for his young children.
There will be time for a national memorial after the funeral.
Holding a public and even "official" funeral is fine. Nothing wrong with at all. But (and I know I am being an etiquette snob here) State Funerals are in a very special class by themselves. Foreign heads of state usually attend. There is an assload of diplomatic protocol and the ceremonial is extreme. I absolutely agree that he was an extraordinary man and true gift of Australia to the world. Some form of public memorial service or even official ceremony is perfectly appropriate. But even royalty who are not crowned heads of state are not entitled to state funerals.