Here you want to go public for emergencies, and private for electives.
I had a mate whose wife was in a car accident and shattered both her knees. When the time came he asked the nurse if they wanted his private insurance details, and she told him no, as that would just mean the whole process would slow down. With public she got seen by doctor and given a room immediately. If they went private they'd have to wait for the paper work to got through, wait for an health fund approved doctor to come around and blah, blah, blah. All for the same treatment. So he let his cover drop because it was useless. A few years later he needed a hip replaced, and he waited forever for a slot in the public system.