If Australia had been a republic in 1975 then surely its constitution would have had to reserve similar powers to its president, in order to be able to break deadlocks in parliament. That they were exercised on behalf of the Queen (of Australia, rather than of the UK) was a political red herring.

The ceremonial and constitutional powers of the monarchy, in the British model at least, are also required under a presidency. The only issue is how the individual exercising those powers is chosen. It is by no means obvious to me that having a politician in the role is necessarily desirable - President Thatcher, anyone? Or President Blair, for that matter?