I do not believe the question of what is core business really goes near the problem. What is the core business of government? Anything and everything that the government decides. Whether the government is achieving key goals in their chosen fields is a more important question. Not where is the money spent, or even how much money, but is it achieving it's goals or even expanding social and material capital?.
I don't think too many people would argue against hospitals being amongst the core business of government. Everyone knows about Qld's waiting list crisis, but from the recent experience of a family member, it would seem that the problem is not just a financial one but one of how the "business" is structured. The family member spent 10 days in one hospital waiting to for a bed to be available in another hospital to have tests done. She was only in hospital for that time because if she went home she would have to wait 12 months for the tests as an outpatient., So she stayed in a high support ward, with little support needs, just constant monitering, while other patients with moderate support needs had to wait for a bed to become available in this ward. The test itself took about six hours including preparation and recovery, but the chain of waiting lists clogged the system like a congested freeway in many other areas of the hospital for 11 days.
Private enterprise would not throw more money at such a problem, they would restructure.
A private business would lose it's customers with service like this. The consequence of this hospital structure is poorer health for Queenslanders.
Whether it is hospitals, the arts or engineering, whatever the business, the culture and structure of the public service is the problem, not the financial strategies.