Having a monarchy, I believe, creates a necessary 'final say' in the most extreme conditions.
Imagine having a game of rugby without a referee. America's referee (the Supreme Court) is so involved in politics that they mayaswell just be some of the players... As Gore Vidal said in the interview with Bob Carr, the Americans have nowhere to turn to externally. The judiciary, the executive, and the parliament are so intertwined that there really is no hope for change.
Having the governer general and the Queen ensures that when our parliament (or High Court) acts up, they can be put back into place. Yes the monarchy is so detatched that they don't know what's going on, but that's actually the point of a foreign, separate power.
I think we're so lucky with our politicians that we sometimes forget that the Queen plays a practical role. The fact that the Queen (and, supposedly, the Governer-General) are outside of the political sphere allows this to function.
If we became a republic, any President would eventually have something to do with the government before his appointment. The head of state should not come from government. Furthermore, it's impossible to establish a hereditary class of leaders -- how would we chose? Personally, I like the fact that the monarchy is hereditary, that way, there is no choice of leader - no campaigns, no corruption, no fuss. Cruel and unusual monarchs are a good price to pay for a detached leader.
Also, there's the fact that the election process of the President would involve the Parliament and the People too much. The Commonwealth monarchy works because they insist on staying out of government matters (some people think it's a bad thing, but they should wake up). The fact that our monarchy spends all day opening parks, making lame speeches, and learning different languages is a good thing. As soon as they get involved in how things are run, they essentially become part of the system that they should be above.
The Commonwealth monarchy is too valuable a tool to get rid of.