The current US Government shutdown which has now lasted for more than a month, is I think an object lesson in why Australia should absolutely not become a republic. Meanwhile, as dysfunctional as Brexit is, it is also an object lesson in why Australia needs to stay as it is.
I think that it is almost certain that the people of Australia want to elect the head of state. The 1999 referendum on the republic, proposed a system which fundamentally didn't alter anything apart from having the Govenor-General elected by a super-majority of parliament. That was turned down; not because there's inherently anything wrong with the idea but because the people of Australia just won't wear it. It must be said that the President of Germany is elected by the Bundestag and electors from the 16 Bundesländer (states) and that the system works very well but that's mainly because Germany has 8 parties and a massive chunk of non-aligned members; Australia on the other hand has a very strong two-party system. Australia probably wants a system similar to Ireland; where the President is elected by the people and holds similar powers to the Governor-General.
The problem with an Australian President as I see it, is that there is no reason in principle to assume that an Australian President would be benign. Australia's political climate, including since before Federation, has always been one akin to a gladiatorial fight to the death; to such a degree that the Parliament of New South Wales has gained the nickname of "The Bear Pit".
Setting aside the fact that in the United States, the President is personally vested with the executive of the nation, the power of veto that the President has is functionally identical to that of the Queen in Britain, the Governors-General of Australia, Canada and New Zealand and the Presidents of Germany, Ireland and France. It is also worth noting that when Japan settled on its Constitution after the Second World War, they cribbed heavily from the best aspects of both Australia and the conventions of New Zealand, and the Emperor of Japan sits over the top.
Having the power of veto, is exactly the reason why this current President is holding the normal functioning of government; holding government workers to ransom without pay until he gets what he wants.
I have complete faith that a President of Australia, who would be elected by the people of Australia, with the same powers as the Governor-General and nominally the same as the President of Ireland, would be a spiteful knave. Although in the United States there is a myth that anyone can be President, both you and I and Blind Freddy can see that that is consistently and demonstrably untrue. The only people who have ever been successful at becoming President of the United States have either been military generals, politicians who were already inside the system, and the current resident who has shown that with sufficient fame and money, actual political skill can be negated and overcome.
That doesn't mean to say that just because the current resident of the White House is a petulant man-child who is willing to hold congress to ransom until he gets his way is an oddity. The grand list of United States presidential vetoes is truly terrifying with more than 1500 pieces of legislation being directly vetoed by the President; with who knows how many more being refused, as is the situation now.
Australian politics is such that knifing Prime Ministers is almost a pastime. That means to say that political climate inside parliament can often be described as "stabby-rip-stab-stab". If the people of Australia were given the power to elect a President then I have no doubt that the weight of that election would create a mandate for that president to do something. Given what I know of Australian politics, that would mean handing the power of veto with mandate to a spiteful knave.
Suppose that Cat Party holds government; so the Cat Party is able to pass the budget on the floor of the House of Representatives via Appropriation Bill No.1 20X5. If they manage to wrangle enough votes in the Senate, which may or may not be hostile, then Appropriation Bill No.1 20X5 would pass to the President. So far so good. What if the President is a member of the Dog Party? Dog Party voters who elected a Dog Party President, have faith that the President not pass Cat Party bills that they don't like. If Appropriation Bill No.1 20X5 does not get Presidential Approval, then the functions of government no longer have a budget or the authorisation to continue; whilst Australian budgets tend to have conditions which would roll forward, unfunded portions of government would have to close; which is exactly what we're seeing in America.
I have no reason to believe that Australia is capable of electing an independent President. I have no reason to believe that if Australia were to elect a President, that we wouldn't get a "stabby-rip-stab-stab" child-President who with the power of veto, would not be a spiteful knave.
This brings me nicely to Brexit. Brexit would from the outset appear to be a tremendous calamity and proof that the system doesn't work. Actually what Brexit proves is that the system works perfectly and it is politics which is broken. The Brexit legislation which was turned down and which is actually the single "worst" turn-down by the House of Commons, is proof that you need a majority of members on the floor of the House to pass legislation. The Tory Party then went on to hold confidence; which coupled with the fact that the budget was previously passed, is also is proof that you need a majority of members on the floor of the House to hold government. If Brexit legislation does manage escape the House of Commons, there will be some adjustment in the Lords no doubt but the power of veto, will almost certainly no be exercised.
The Queen who sits outside of all of this and who is a non-elected person, has seen 14 Prime Ministers in her time, of both major political tribes, passes everything put to her. The last time that Royal Assent was refused was back in 1708. In Australia, I actually can not think of a single piece of legislation which the Governor-General turned down. In effect, the legislature which is made up of elected members is where the power actually lies and because nobody really knows what powers the Governor-General has, they are loathe to use them. I like Australia's benign Governor-General because although they are not elected, nobody really cares about them. A benign nobody is preferable to a spiteful knave.