But I think the distinction is still important. For example I think there’s a good case for limiting democratically elected politicians influence over the economy, by enshrining private property rights and the independence of the central bank in the constitution (as e.g. Sweden has). How would you come to that conclusion if you think it’s democracy itself that allocates resources efficiently?
Democracy acts as a circuit-breaker for entrenched inefficient resource allocation. For instance, if there was no usable legislative process, but only an executive and a precedent-based judicial system, markets would generally be free, but this would allow the intergenerational accumulation of capital/wealth until some people have almost total power, and some people are essentially serfs. But in democracy, the mere potential for the mass of people to vote themselves wealth limits the incentives to aggressively accumulate capital/wealth. For instance, if the rich want their property rights to be respected, they need to make sure that the mass of the country has enough property to also want protection. And if that potential doesn't work, it can actually be used and new taxes and rules can be introduced that distribute wealth more efficiently.
As you can see, some might not agree with your conclusion that "there's a good case for limiting democratically elected politicians['] influence over the economy, by enshrining private property rights and the independence of the central bank in the constitution". By asking the question you did, you first presumed that democracy was inefficient and on that basis drew a conclusion, but then asked how a person who thought democracy improves efficiency would argue for the conclusion whose premises they disagree with.
But in any case, I wonder if your example is not so strong - after all, the independent CB was only enabled by democractically-elected representatives voting for it.