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Structuring healthcare as an insurance doesn't work because pretty much everyone needs it at some point. It needs to be structured like a tax, because the rate at which you pay in has no correspondance with how much you need to get back out, but in aggregate the total sum paid matches the total care provided.you've just outlined a money-losing system run by the government.
the reason private health insurance systems fail is not merely because everyone needs it at some point. it's because everyone uses much more of it than they've paid for with their premiums. (which is why, pre-Obamacare, insurance companies used to reject applicants with pre-existing conditions and why they put lifetime caps on benefits paid.)
but a government run single-payer system is not magic. the extra cash must come from somewhere.
one way a single-payer system could obtain the additional funds needed is by taking cash from some other government source (which is exactly what Obamacare does when it provides refundable tax credits to help low income people pay their otherwise unaffordable health insurance premiums.) but there's no end to the amount of money that could require.
another thing the government can do is limit services to patients, i.e. take a certain amount of decision making authority, by law, away from individuals, especially old individuals who are very sick. ("i'm sorry, but we're not paying for that new chemotherapy. it's hospice care for you.")
in addition, a single-payer system can also uniformly limit the prices health care providers, pharma companies, hospitals, etc can charge. (i.e take some decision making authority away from that sector of the economy, again, by law.)