The provisions in the Constitution that the Governor-General is appointed by the Queen as her representative and at her pleasure are archaic even if it is acknowledged that the powers are used only on the advice of the Prime Minister of the day. It is demeaning to our sense of sovereignty. As the present federal electoral cycle is a limit of three years, any president will span at least more than one parliament in five years or probably two if for seven years. Even 'ceremonial' residents, such as Mary Robinson in Ireland, were political players. A president cannot preside for life, so must have a fixed term.
Most republicans suggest five years or seven years.
In most republics, removal of the Head of State is a parliamentary process. This may however involve one or both Houses of Parliament, the Chief justice in some capacity, a Constitutional Court or popular referendum. Thus none of these 'solutions' is nearly as good as our present system. A president must either have some individual or body with the power to remove him or her. Most republicans say this should be the Houses of Parliament voting by a two thirds majority.