Data shows otherwise, source: https://ourworldindata.org/grapher/life-expectancy-vs-health-expenditure
And the funding model (public vs private) is orthogonal to using or not using the price signals - nothing prevents you from using market information to determine best value for money, which is precisely what most EU providers do, for sure UK NHS and PL NFZ (out of those I know). Both have massive statistical divisions who constantly analyse available market offerings both for drugs and service providers and change them in order to optimise their costs.
That's also precisely what US private providers do all the time, except they are optimising for the FINANCIAL profit of their shareholders, while NHS/NFZ optimise for the health benefit of their patients.
