Now apply that logic to healthcare. Single payers are able to get much better deals.

Not really, no. Medicare wildly overpays for everything compared to the rest of the world.

Wouldn't a more apt comparison be between Medicare and non-government insurers in the US?

There isn't really any way to make that comparison because Medicare provides a different type of benefits to a different subset of the population. Private insurers don't offer policies to the elderly that cover the things Medicare covers because those things are already covered by Medicare.

People sometimes try to measure things like "overhead" but that also doesn't really work. If one entity pays $10M in claims but half of them are unknowingly fraudulent and another spends $1M to prevent the fraud, the first one is paying $10M with 0% overhead for fraud detection while the other is paying $5M with 20% overhead for fraud detection. Comparing the percentages implies the second one has "higher overhead" but it's also the one paying $6M instead of $10M.

And the same thing bleeds into the prices for specific services. A provider will happily give you a "20% discount" on the price if you don't bother to check that half their claims are fake. Then you're actually paying them 60% more while the books show them charging you 20% less per procedure.

On top of that, the market for many medical services in the US is extremely captured. The majority of states still have Certificate of Need laws, which are literally laws prohibiting new providers from entering the market if the local government (i.e. the incumbents) think the result would be "too much" competition. The AMA consistently lobbies to limit the number of medical residency slots and sustain a doctor shortage, and doctors who immigrate from other countries, even licensed physicians in countries with first rate healthcare and medical education systems, can't practice in the US without doing a US residency, which in turn consumes one of the residency slots and prevents immigration from alleviating the shortage whatsoever.

The premise is that competitive markets result in lower prices, not that uncompetitive markets do.

Medicare covers only a minority of the population but (and?) relies heavily on the general US private healthcare system. At minimum, Medicare cost is affected by high drug prices (perpetuated by lobbying-influenced government avoidance of price controls), relatively high coverage denials (which Congress and the executive branch refuse to regulate), and lack of vision or dental coverage unless people use "Medicare" Advantage (which currently gives private insurers too much leeway to raise costs for the government without a corresponding healthcare benefit [1]).

Anyway, I think parent comment to yours is using single-payer in the sense that implies universal healthcare [2].

[1] https://www.kff.org/medicare/how-medicare-pays-medicare-adva...

[2] https://en.wikipedia.org/wiki/Single-payer_healthcare

Drug prices are less than 10% of all US health care spending. Drive all drug prices to zero dollars and we get a grocery store circular discount on total health costs.