Do you consider democracy more a preference discovery machine, that is good at finding the preferences of the population and thus gives insight into the preferences of agents. Or do you think democracy is more a mechanism of aggregating information in a wisdom of the crowds way where the input of lots and lots of people leads to good outputs.

Of the two I'd say something like 20:1 preference discovery over wisdom of the crowds.

This is me hypothesizing on the fly: every governing system puts day to day control in the hands of a tiny minority, and an even smaller group of people who have the time, resources, and legitimacy to manage that tiny minority. It is inevitable that the decisions will not please everyone, and in some trivial sense, it is inevitable that the decisions will not please anyone, even the single most powerful government leader, who must still find himself constrained by human nature and other realities.

Of the choices I am aware of, a republic with principles such as (near) universal citizenship and (near) universal suffrage results in the greatest breadth of consideration of the preferences of the population. If I had a choice between getting rid of the bill of rights or getting rid of universal suffrage, I would toss suffrage first. Voting is just ONE tiny way that the information flows up from the bottom to the top. Voting seems to put a nice catch-all loop around the whole process, but I'd want to keep the day to day guts, enabled by strong concepts of nearly universal rights, than to lose the day to day and have a mass of lied-to and oppressed people voting once in a while.

The problem with all alternatives I usually think of are they move MORE power into the day-to-day runners, which are necessarily a tiny minority. Even in a well functioning republic, it is hard to distinguish whether the legislators all get rich because the skills that make you rich are what make you electable or because they vote with a bias towards their own interests, at least every once in a while on the margin, or because they exploit inside information, or because they are handed wealth by those who wish to "befriend" them and influence them.

Of course in a republic it will always be all of the above, but the democracy component will be constantly throwing stumbling blocks in the way of the specially good results for the gov't insiders. FOIA, laws against secret donations, laws against many kinds of "gifts," laws about disclosure.

And then after democracy has done what it can to systematize limits on "natural self-interest" or "corruption," depending on how you like to talk about it, we get the election, where as long as we are still relatively free and secure in our rights, those incumbents who can be attacked as more selfish than helpful will be. So the interests of the governed will have MANY mechanisms for setting an environment where the interests of the governing are tightly limited.

So in detail, democracy provides a gigantic amount of discovery of preference in a deep way, with strong popular rights to information and to speak combining with contested elections to produce a system which almost looks like it is policing itself!

Compared to the essence of corruption control, the additional fact that democracy nearly by definition enfranchises a larger fraction of the population than do many other systems you mention (monarchists, religious or any other form of dictatorship, marxist/communist where traditionally choice is left to or at least heavily constrained by "ideological experts").

So yes, a preference discovery machine both broad and deep.

As to wisdom of crowds, its hard to distinguish between preference and wisdom. Most of what you use a prediction market for are predictions of what crowds will do. Obviously when predicting elections, but even when predicting future technological developments, the mechanism of prediction market seems to be to attract more information into the open by providing a mechanism for the small number of people with super information to profit from publicizing at least their opinion. There is a component, probably, where gov't seeks to placate people by doing X, but Y is actually better at placating them and democracy can in a sense help reveal that fact. (notice with propositions on California ballot, I learned that most Californians are de facto willing to pay $20-$100 million extra to execute a small number of people than to use those resources in ways that might prevent crime, a result which surprises me.)

So preferences, definitely.