The British monarchy has nothing to do with core weaknesses in the Australian constitution. The Queen is not the problem. She did not sack an Australian prime minister in   November 1975. An Australian governor-general did, contrary to the conventions that the Queen dutifully obeys. 
Yet the only change that Australian Republican Movement chairman Peter FitzSimons wants is to remove any role for the Queen. This would still let a future governor-general or president veto legislation passed by the Parliament and sack a prime minister with a majority in the House of Representatives.
Ever since the Whitlam government abolished appeals to the Privy Council, the residual symbolic ties to Britain don't really threaten Australian sovereignty. But the Turnbull government has endorsed provisions in the Trans-Pacific Partnership allowing foreign companies that have lost in the Australian High Court to win potentially large damages by appealing, in effect, to overseas tribunals that aren't even courts.
Unlike the official republican movement, NSW Solicitor-General Michael Sexton and others have explained that a republic doesn't need a governor-general or president. For example, bills could be become law once passed by Parliament, without royal or presidential assent. It's an attractive option for very small "r" republicans, particularly if state governors also disappeared.