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Nobody is saying universal (single payer) systems are without their faults, but no one is denied health care. Budgets are trimmed, waste is slashed, and taxes go up, but health care still remains a fundamental right. Under a free market system, you are at the mercy of a corporation. You have no guarantees so it is the policy holders who get pushed out of the system.
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In some cases you are denied health care in single payer system. See most elective surgeries in single payer countries. You could wait years to get a hip replacement and could die before received one. I suppose we can say you were in the queue so you weren’t denied but effectively you were. You didn’t get the surgery you wanted/needed for an improved quality of life before your life ended. The same is true under a capitalistic system if you don’t have the money for a hip replacement you don’t get one either.

Single payer is essentially a price control system. Since there is only one payer of services, the single payer sets the price for each service. You could implement price controls without single payer. We all know what tends to happen in price controlled systems. There usually ends up being shortages because the price is artificially lowered.