I'm not sure I agree with you. As you say, roads and bridges aren't given to a single person, they're provided for all to use. And police and fire provide a public service as well. But at the same time, all of these, while benefitting society, are also provided to individuals, which you seem to be against. Take for example the fire department - if my house is on fire, the fire department comes to put out the fire. Is this aiding an individual? I'd say arguably yes, while at the same time as aiding society. Sometimes the two go hand in hand. Roads and bridges are the same way... they're a public infrastructure, but used by individuals, for their individual needs (and at different rates of usage, I should add).
So along these same lines, healthcare is arguably a public service. I'm not saying people shouldn't have to pay for it... but like many of the other public services the government provides which most people will likely need and use at one point or another in their lives (roads, bridges, fire, police), the government has the responsibility to provide healthcare as well.
So whats a 'basic level of healthcare'? I'll use France as an example (I live there so I have experience in the system). In France, everybody is automatically covered under 'Security Social' or 'Secu' for short. Essentially, Secu is a system of reimbursement (like insurance) for health care needs. For most things, Secu covers about 60% of the cost, but can cover up to 100% depending on the procedure. However, Secu doesn't cover dental visits, corrective eyewear, or most prescriptions. If you want more coverage on medical procedures, plus dental, presription and eyewear coverage, then you have the option of getting a 'top-up' private insurance (mine for example costs 30€/month).
So as an example, when I go see a general practitioner, I pay 23€ (the cost of a visit at any public general practitioner). If I had just Secu, I would get reimbursed 60% of that about a week later, so the final cost to me is around 9€. With my 'top-up' insurance, I pay 0.