The Governor General tends to use none of the powers you're talking about being a threat. Suggesting a president would use them where the GG didn't is probably a bit outlandish.There is actually strong evidence that a republican government would fundamentally alter the balance of Australian political power and anyone who tells you that a republic would not be detrimental to the governing of Australia is talking out of their ass.
Time for a lesson in constitutionalism:
The Governor General is head of the executive according to the Constitution. They can appoint and dismiss ministers as they please. They can also call elections under certain circumstances. Obviously in practice these decisions are made on advice by the Prime Minister who can also advise the Queen to dismiss the GG.
Because the Crown itself is trusted and normally only operates on the advice of democratically elected advisers, the Governor-General has been entrusted with vast powers. These include the command of the armed forces, the power to call and prorogue Parliament and the power to choose the Ministry, whether the ministers be in Parliament or not. The brake on undue or improper exercise of the Governor-General's power is the threat of his or her instant recall by the Queen on the advice of the Prime Minister.
But just imagine that we abolished the monarchy and became a republic. Our GG would be replaced with a President who we shall say is directly elected by the people but otherwise fulfils the same functions as the GG does now: they represent Australian executive power, they are in control of the military and so on.
What happens when a President chooses to use their reserve powers? On whose authority do they use them? What if the use of these powers is a flagrant breach of convention and seriously destabilises the government? Can they somehow be removed from office? Should they be dismissed given that they will have an electoral mandate? Whose electoral mandate is superior: theirs, or the governments? Does the government even have the authority to do this? If the government does somehow undermine the office of the President, how will that be resolved given that they are constitutionally Australia's head of state? Doesn't that represent a gross overreach of government power? There are no clear answers to these questions because the elegant balance of power between Parliament and the executive/head of state that we now have would be utterly destroyed.
These are just a few important questions that can be raised if one adopts the 'minimalist republican' approach that is very clearly totally incompatible with the Constitution as it stands. Imo, to successfully implement a republican government would require a major redesign of the Constitution, something night on impossible and which should not be done given that it is the fundamental law of our land.
With this in mind, the fact that the constitutional monarchy has provided us with decades upon decades of stable, democratic government, is cheap to maintain (and the costs of transitioning to a republican government would be enormous), has adapted well so far to shifting cultural values, and provides a tangible link to the institutions that all Australians identify with, means that there is no logical or rational reason why this country should become a republic.
Legally our links to Britain have slowed down enormously the development of our legal system. Hardly a good thing. Anyway, Britain's influence on us is less than America, whom we have no monarchical ties to.