The airline could quite easily operate under a foreign AOC and do what they do but they choose to fly under Aussie regulations unlike a lot of their competitors
In way of background, the former Air Nauru was very close to being shut down by the FAA and CASA because of the lack of a proper civil aviation authority and staff on Nauru. One man and his Frigate Bird does not make a CAA.
The operation was rescued by one former Air nauru chief pilot then a consultant to the Government of Nauru, who made it clear the only way the airline could continue flying and satisfy the FAA and CASA was to come under the Australian AOC system and have its aircraft registered as VH-. This was essential for the survival of the operation otherwise the 737's may as well just flown their hours doing circuits and landings on Nauru.
This FAA interest stemmed from the Korean Air B747 crash at Guam, where the FAA were shaken by the circumstances surrounding that crash on US Trust territory. This included questions of Korean pilot competency, fatigue and a host of relevant flight safety issues.
Very quickly, FAA inspectors looked at foreign operators (including Air Nauru) landing at FAA jurisdiction airports such as Guam, saipan, Chuuk, Pohnpei, Majuro etc. Audits revealed serious problems with some foreign operators. While Air Nauru has had an excellent safety record since its inception in the early Seventies this was primarily due to mainly Australian crews including Ansett and Air New Zealand contracted engineering staff and the dedication of the Nauruan government to insisting on first class servicing.
After a disastrous strike in 1988 by the fledgling Nauru Air Pilots Assocation, the then President of Nauru, hired pilots from Indian Airlines and some Eastern European airlines to replace sacked Australian pilots. Immediately the airline operational standards dropped significantly. Eventually the Indian pilots left to be replaced by Australian crews who themselves had lost their own jobs during the 1989 AFAP punch-up.
Operational standards then improved and at the time of the Korean Air accident it was back to its former good standard but still not helped by lack of a competent local DCA. The move to Australian AOC and VH- registration was expensive but successful although the airline financial situation was dire as always with huge government subsidies until the phosphate money ran out to just a trickle.
To say the airline could operate easily under a foreign AOC is drawing a very long bow. The new Air Nauru (Our Airline) probably had little choice but to stick with its Australian AOC and attached operational standards, otherwise face intense scrutiny from Australian authorities and the FAA if and when the operation expanded into some of the former destinations controlled by or certainly influenced by the FAA. The comments above are a little generalised but not far wrong certainly in the history of the move to VH- registration of their aircraft.