While I generally agree with your sentiment, I think the health insurance example is not a good one. There are several factors that are wrong with it, for one people with insurance have advantages over people without, so you are creating a system where your access to healthcare depends on how much you earn. Secondly, it essentially creates an artifical market for insurances with pretty much guaranteed customers, the insurances don't even need to offer that much over the state system because it's cheaper anyway. If you want have the universal health care not be as loaded, simply raise income/wealth/other taxes and use that funding to prop up health care. That's a much fairer solution.
I'm not sure I understand. The Australian system is weird that it isn't a insurance based system (like e.g. Germany), but they give tax insentives to high earners to get insurance. However those insurances don't pay for the basic stuff paid by the medicare system, but for extra things, for example you get faster treatment for elective surgery.
