If and when any country becomes a republic, IMO dual citizenship availability is should be withdrawn immediately...
Republics also allow dual citizenship, Italy for example is a republic and yet allows its citizens to hold dual or multiple citizenships, from other republics as well!
Until the 1970s (if my memory serves me correctly) Australians weren't allowed to hold dual citizenship but that changed as we became more multicultural.
Until 1972 Australians were British subjects and Australian citizens and travelled on British passports, after 1972 Australian passports were issued. The Whitlam government brought it in, prior to 1949 all Australians were just British subjects, there was no Australian citizenship. Dual citizenship was brought in as other countries allowed their citizens to keep their citizenship when they became Australian citizens. Beforehand certain countries required you to renounce your old citizenship if you became an Australian citizen. This was the case with Italy as I counsel frustrated young people (who want to work overseas) who have Italian born grandparents and parents and find they can't get an Italian passport (EU one) through descent as prior to 1992 the Italian government required that you renounce your Italian citizenship if you took on Australian. Their parents and grandparents became proud Australians on paper and their descendents lost forever the right to dual citizenship, those that became Australian citizens after 1992( due to change in Italian law, not Australian) can get dual citizenship. The British government always has allowed for dual citizenship and many young people then qualify and get their EU passport by having just one British born grandparent.
Most Australians who have dual citizenship don't hold British and Australian citizenship but that of their homelands.
I think you'll find that for adult Australians who have dual, one of those citizenships is Australian, simply if they work overseas for extended periods they risk loosing their permanent residence status in Australia. I have dual citizenship, one being Australian, one EU (not British) and I qualify for another throught descent. I could have also qualified for permanent residence status for another country due to time spent living and working there, I'm not that unusual when looking at young mobile professionals, thanks to Australia's multicultural nature being able to move and work worldwide isn't an unusual situation.
I have no British descent or history and am quite happy for Australian to stay a Consitutional Monarchy and be spared totally unnecessary expense. In all honesty it would make no difference to have a president or Governor-General except for the cost. First the cost of having the public debate and referendum, the 1999 one cost well over a million dollars, ( as far as I'm concerned money better spent on education, rural health care, road, disablity services, all of which actually make a real difference in people's lives) If Australia becomes a republic the model that seems to have the most support is the elected president, so then there's the cost of a political campaign by the candidates, all funded with government money. (which again could be better spent) Then finally when the president retires, he or she receives a pension for life, along with other perks, more money! The office of the Italian president (purely ceremonial) currently costs more to run than the monarchy does in the UK, it's a good comparison as the population is about the same, Italians are also paying for the pensions of at least 3 former presidents who are still alive.
For a figurehead Head of State, currently money is not spent on campaigns, the monarch inherits, the governors-general are appointed, the role is ceremonial, spend money on things that actually affect Australians in their daily lives.