Or the opposite: Have the state take care of routine preventative care (exams, age-related tests, etc), the patient would have a high-deductible policy for stuf

Or the opposite: Have the state take care of routine preventative care (exams, age-related tests, etc), the patient would have a high-deductible policy for stuff like cancer, auto-accidents, etc. And everything else people pay on a fee-for-service basis. Society benefits because we're paying to ensure a basic level of health for everyone and not the current system of inflated prices. The insurance industry gets to stay in business (which they wouldn't under a government run single-payer system), and patients benefit because they get to shop for a provider based on outcomes and price, for everything that isn't catastrophic.