How do you feel about Denmark's constitutional
monarchy? Do you admire it?

Australia must become a republic when the Queen passes on. The thought of having her disastrous son, as King of Australia is totally repugnant, He was the man who was already having an affair when he married Princess Dianna . Remeber her famous words " There are three of us in this marriage" . We do not want him. We must become a republic.

AS for who is head of state? We can elect a governor general as one option .We are unlikely to make the same mistake that the American did in electing Donald Trump, as we do not have the same divisive party system as does the United States . We also do not have the same history of a war of independence or of an internal civil war . We are a less divise people where "I am as good as the next man, mate!"

Australia will win out even more if it becomes a republic

As far as I can tell no proposal has been put forward
that betters what we already have. It stands to reason
that if we're going to remove what we have we've got
to replace it with a better system which is more
effective in keeping politicians from abusing their
position.

Elect a President and you install another power centre.
That is the major problem and the cause of countless revolutions around the world.
Paul's alternative is a possible but means that someone who never was
anybody and is probably an unknown gets the job. Q ? Whose puppet ?
The Govenor General can dismiss the government if he believes the
government is acting illegally; note Kerr consulted a High Court judge
if I remember correctly. The government was borrowing without
Executive Council permission from Kemlani an Arab weapons dealer.

Why have a head-of-state at all? Isn't a head of government enough? Do we need another snout at the public trough?

The Governor-General is the Queen's representative in
Australia. If we want to change this - then we need to
become a Republic. We are an independent nation except for
the fact that -
we share a monarchy with the UK and many other countries
including Canada and New Zealand. The Queen's powers are
delegated to the Governor General by the Constitution.
But by convention the Governor General can act only on the
advice of the elected government.