A republic is definitely important to Australian independence from Britain. With a republic, every person in that country in some form or fashion, can have a say in what their government does and how their government does it. People can truly play a part in their government, and that will allow the transition to a free country to happen quickly.
Since the beginning of Australia they have been dependent Britain. I believe one of the steps of claiming Independence of Britain is to differentiate themselves form Britain entirely. This is not the only thing they need to do to gain independence from Britain, but i believe it is a necessary step.
To say Yes to this, is to suggest that somehow Australia isn't a free and independent country, something which it most certainly is and has been for some time. Australia could have a president tomorrow and not an awful lot would change, the Royals would still visit as they do many non-commonwealth Nations; Australia would still have a close relationship with Britain and you're never going to undo history which will always shape our relationship.