Australia has a combination of public and private health insurance, and they both work well together. The public health options provide the safety net, while the private health insurance is optional.
Where the private option makes sense is if you want to go to specific private hospitals, or if you have elective surgery (the classic example being a knee reconstruction for sports injury) and you don't want to be in a queue behind people waiting for public hospital beds for more serious conditions like heart surgery and so on.
My dad in Australia had open heart surgery 2 years ago, and is doing very well. His cost for the entire procedure? $0 and this was done by one of the very best heart surgeons in the country. He has private health insurance, but elects to go to public hospitals, which have excellent surgeons committed to the best care, because he's a patriotic sort and he's paid into the public health system through taxes for his entire life.
Meanwhile I pay > $3k per-month for not even top tier care in Upstate New York for myself, my wife and my 10 year old daughter, with no serious pre-existing conditions, and I have absolutely no guarantee that any surgery or anything that any of us need in the future will be even covered, even if my primary physician says it's medically necessary.
The rest of the world would do well to study how the combination of public and private health insurance is done in Australia.