I found a few resources for you related to your question.
The first is an opinion piece from Independent Australia. The author argues that an independent republic would serve to complete Australia's journey to independence. As Australia has "gradually moved away from Britain," it has established itself as a unique and self-sustaining nation. However, the current constitution still contains language that discriminates against Aboriginals and implies that privileged British royalty are "above" Australian citizens. Being an independent republic would allow Australia to recognize its history and heritage, demonstrate equality, and determine decisions for the people by the people.
The second article details the decisions that led to Scotland's proposal for independence. Although the movement was not ultimately successful, it seems like the quest for independence gave Scotland an autonomous voice that was recognized around the globe. As rational human beings, it seems like we balk at the idea of outside restrictions, and blossom when we challenge those constraints.
Even if Australia does not eventually become a republic, the process of exploring true democratic independence can only serve to raise awareness about hope, freedom, and equal rights for all.