The Australian system is far from perfect but does come some way to solving this. It is effectively 2 tiered:
1) Public Medicare, the publicly funded system available to everybody.
2) Private insurance, paid for by the individual and tailored to their needs.
To pay for the public system everybody pays a Medicare levy as part of their income tax. The levy is coarsely means tested. If you have private insurance then this levy is reduced somewhat.
The real critical differentiation between the public (Medicare) and private offerings are that any services deemed as elective in nature (ie, non life threatening conditions such as surgeries to treat injuries, etc) can be completed by the public system but there is a wait list which can often be quite long. If you have private insurance then you can get in very quickly with the surgeon you want.
Having private in no way restricts your usage of the public system so there is no downside to having private except that it costs more. There are also fee structures in place to encourage people to take up private when they're younger and the saving on the medicare levy.
As I said, it's far from perfect, but the 2 tiers offer a high level of customized cover for people willing and able to afford it while offering an acceptable level of cover for those who can't.
