That is a complete misnoma Tom monarchy plays no role in our system since it has no real power excepting to sack a government. Such power as there is is held by a representative of monarchy appointed by the government. The executive branch is the Governor-General and Prime Minister operating in the Executive Council so the Governor-General takes the advice of the Prime Minister
Only once has this been challenged. That was in 1975
The laws of Australia are decided by its Parliament and given assent by the Executive Council. The final authority on the legality and constitutionality of those laws is the High Court of Australia. It is unlikely that an Australian republic would be formed on any materially different structure and attempts to form a republic based on direct election of a president have already failed. The republic issue is unlikely to be canvassed again in the reign of Elizabeth II. To do anything requires amendment to the Constitution and that has proved extemely difficult to do. in any case Tom what is your President other than an elected monarch?
The reality is, Tom, that our democracy is far more representative than your own, with the Executive Branch, represented by the Prime Minister, facing questions in the House on every sitting day. There are no staples to bills, there might be deals but it will be in seperate legislation which must itself face the House, or it waits for the budget process