There are some "costs" problems with health "safety net" concept. Free general health care can lead to abuses (over-use by patients and over-service by doctors).
Health insurance (private and government subsidised) increases complexity of patient identification, careful description of complaints (wrong words may deem patients uncovered), and billings. (More effort is spent on administration rather than on treatment and patient care.)
Shared responsibility seems a middle ground. Here the service costs are paid by patients and health (insurance) funds. Because patients have to pay, over-use is minimised. Because patients have to pay, over-service is not always agreeable to patients.
Self-care and community-care models can be formulated as cultural solutions to promote self-care know-how and to create community care facilities as sufficient need arises. In short, a culture of good health and well-being for one and for all in any community is the aim.