I agree. if I were to design a constitution, I would abolish the office of Prime Minister and make the President head of government as well as head of state, and force all cabinet ministers to be chosen from outside the Parliament. That sounds a lot like the US republican model eh?
The Commonwealth Games is a different matter. From what I watched of the Games last night, I see that the organizers treat the Queen as the dignatory representing the Commonwealth as an organization rather than Australia's HOS - apart from God save the Queen and birthday song session I didn't see anything really monarchist there.
For all practical purposes, the Queen ISN'T our HoS. That's the role of the Governor-General.
Britain has NO say over what happens in Australia. Another plus for the system!