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.)
First, how it is paid. It's text book socialism. People pay into it based on ability and receive benefits based on need. It doesn't run at a loss.
Second, the level of care. Basically the government puts a lower bound on health care and uses taxation to do it. You're guaranteed a basic level of care regardless of your situation. You can opt for private insurance that stacks on top of that. A basic level of treatment is covered by the government, but anything experimental or exorbitant you will likely have to pay yourself, unless you have the aforementioned gold-plated insurance. Insurance is much cheaper because it only needs to cover unusual care, but of course I do pay a lot of taxes.
It's not different from a private system, because you can't receive unlimited care, but unlike a private system you're always guaranteed a minimum care.