A Question for Single-Payer Proponents

I have a question for anyone who believes that a single-payer health-care system (where the single payer is government) will reduce the quality-adjusted cost of health-care: will a single-payer pet-food system (where the single-payer for pet food is government) reduce the quality-adjusted cost of pet food?  That is, under a single-payer pet-food system (with government as the single payer) will consumers be better supplied, at a lower cost, with pet food than consumers are supplied today with a free market, myriad-payer system for the provision of pet food?

If not, why would a single-payer system for health-care reduce the quality-adjusted cost of health-care?  But if so, why not socialize the entire American economy given that a single-payer system (with government as the single payer) would clearly deserve the presumption of being a superior method of economic organization than myriad-payer free markets?

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Although I've not seen such a question asked before, I have little doubt that I'm not the first person to ask it.