I ask because I have met people from Australia on my travels and many of the wealthier ones all have private insurance indicating that the Australian system just wasn't up to their liking.
It's really, really hard to compare different single payer/universal healthcare systems. Medibank (now Medicare) here has evolved far from its original scope.
Hospitals are operated at state level while our single payer system is federal. We also have a subsidised pharmaceutical scheme. Both our single payer system and our pharmaceutical benefits scheme have annual safety nets which reduce your out of pocket costs once you hit a certain threshold.
Health insurance here reduces the gap between what providers charge and the rebate given by the government. The government is the primary insurer. People pay a levy through their taxes and the amount of the levy depends on their income. The government wants people to have private health insurance so the levy is a higher percentage for higher income earners.
Health insurance here has become more restrictive over time. "In network" never used to be a thing here and neither did pre-approval. A lot of low and moderate income earners have abandoned private health insurance as it's become more expensive and more restrictive and are choosing to rely on the public system.
We do preventative care extremely well and we do critical care really well. A year of medications for cancer treatment cost me less than $200 and my only other expenses were for parking on the rare occasion I didn't organise a permit and for food when I was over the hospital menu after multiple hospitalisations. Five years on, I do not pay for any of my surveillance imaging or consultations with my specialists.
It is the public hospitals here which handle trauma and other complex cases. Private hospitals have expanded their scope over the last few years so many routine surgeries are performed there for those with private health insurance, which frees up beds and other resources in the public system.
Some things are also not covered by the government here. Sometimes people take out private health insurance so they get some of their dental and optical expenses covered (coverage for a whole lot of things is under "extras" but the rebates aren't always great). Having private health insurance is a big advantage for people who may need inpatient psychiatric care or day patient procedures. While private health insurance does have waiting periods, the longest the waiting period for coverage for certain things can be is 12 months - it is shorter for many.
My grandson is having his adenoids removed and grommets put in next week. My daughter didn't want to wait so it is being done privately. The cost is $5000 less whatever the government rebates are for the surgeon and the anaesthetist. The government pays no rebate for hospital costs for day surgery. He will need his tonsils removed next year but their private health insurance will have kicked in by then and that will cover the hospital costs as well as some of the gap for the surgeon and anaesthetist.