You actually pay more taxes if you do not choose the private option. Though private insurance is generally more expensive. Over the years since it was introduced, there was an exodus of sorts from public to private due to this incentive. It has recently started to turn as insurance companies have been drawing more profit than the value they provide.

It helps that the standard of care in Australia, for significant or life-threatening health situations at least, is generally as good if not better in the public system. For everything else, insurance or otherwise, the only way to get quality healthcare is money and contacts.

As a system of encouraging higher spends on healthcare, private insurance works.. The only issue is none of the profits of that system goes towards better healthcare. It is just more damn rent.