I wouldn't want the Australian system here.
My uncle developed skin cancer after working outdoors on the railway building programme for years with no protection from the sun (he was a pure redhead). He was only covered for the removal of the cancer, not for making him look 'normal', as it doesn't cover 'cosmetic surgery'. As he'd been a contracting worker, he didn't have a reliable income to pay for insurance (and there's always at least an element of 'I'm strong, fit, why would I need insurance?' - by the time skin cancer was being realised to be a major issue, it was around 25 years too late for him). As it was a slow growing cancer, he waited what he termed 'a few years' because he wouldn't be able to afford the medication recommended until he qualified for the extra help. But that decision to wait meant the cancers still progressed. He was traumatised by what he saw after his surgery.
The NHS recognises that at times, cosmetic surgeries are medically necessary and provides them free, such as with cancer.
Medicare does not provide benefits for the following:
most dental examinations and treatment;
most physiotherapy, occupational therapy, speech therapy, eye therapy, chiropractic services, podiatry or psychology services;
acupuncture (unless part of a doctor's consultation);
glasses and contact lenses;
hearing aids and other appliances; and
home nursing.
Whilst not all of those are provided by the NHS, some are - ones I personally rely upon and could never afford privately, by the way - and some are at least provided at a significantly lower cost.
Even emergency ambulances aren't covered unless under specific circumstances and in particular areas. Can you imagine needing an ambulance but knowing you can't afford it - or waking up in hospital and realising you're going to get a massive bill for being taken there?
If somebody has a severe mental illness, Medicare only allows them to be ill and in hospital for 190 days in their lifetime. They still have to copay for each admission. Mental health screening is only covered once a year - so if you're depressed, you'd better tell the doctor first time, as if they need to ask you twice, it's going to cost you, as it will if you've got an alcohol problem, including when you're pregnant. Can't afford a second screening? Oh well, Fetal Alcohol Syndrome awaits. And if you don't need to be admitted, it's going to cost you.
Looking at all the things that are covered and not covered and what is part covered and what needs to be claimed back and that the cover varies from one place to another, it looks to be considerably more complicated and bureaucratic than the NHS is. But insurers and private hospitals/doctors/etc make a fuckton from it, so I suppose they're happy with it, as will be the people who can afford insurance that actually covers them for their conditions.
I'd far rather have the NHS than that. I can get hit by a bus and not charged to be scraped off the road and taken to hospital. I won't have a bill for being admitted. I won't have a bill for developing PTSD. I won't have a bill for physio or surgery with the intention of improving the appearance of injuries. I won't have a bill if my injuries mean I need speech therapy. I won't need to get a credit card out to cover doctor's bills in the hope that I'll be able to claim it back later. And I won't be given distressing news and decisions to make that include an estimate of how much each option costs and have to decide on that basis.
All I have to do now is buy a pre payment certificate for any prescriptions and the most expensive medication, physio, podiatry and other services I need are provided absolutely free through the hospital. It costs me nothing for the GP to check in on my mental health whilst I'm there for something else. They can quite sensibly doublecheck what I eat/drink/etc just in passing without it having to be charged.
The people who think the Australian system is great are the ones who can afford insurance and all the extras. Ask the people who can't afford it and you'll get a completely different story.