Yes, we do need an Executive - if we don't then the system of government we have wouldn't work. Our Constitution brought into existence a system of representative government in which those who exercise legislative and executive power are directly chosen by the people. We might not have an absolute separation of powers, since the Ministers and PM are the Executives and also the Legislature, but we still have possibly one of the best system of government in the world. The Governor-General, though the repository of executive power, does not personally exercise that power, being bound to act with the advice of the Executive Council.
If you'd want to get rid of a constitutional monarchy, we'd obviously need a replacement to the Governor-General, so in essence a President. If you support a republic, would you support a shift towards an American type republic system of government, where the Executive is elected?