Actually, it doesn't. A perfect monarchy is much better at settling conflicts than a perfect democracy. That's the appeal of the Benevolent Dictator approach: there's only one Decider and he Decides things and everyone is happy with his Decisions. It is a secularized faith system, with King Solomon deciding to cut babies in half to provide some Zen satori moment.
A democracy has no such mechanism. To the extent that we do (see our obsession with elections of public servants), it is a measure of its imperfection. The fact that we permit so much weight and meaning to go into who becomes POTUS or who becomes mayor or the the senator of New Mexico or whatever: these are indicators of the weakness of our democracy. They're fracture points.
> But it might have other problems, for example minorities will probably be disadvantaged. More able people will be exploited by the less able people, too.
One of the advantages of democracy is that egalitarianism (one of my values; as a liberal, I'm individualist, egalitarian, melioristic, and universalist) is possible. It's not possible in any other system we've come up with so far, not even an anarchistic lack of system. In an anarchy, government will naturally arise. In a non-democratic system, the government is necessarily in a power imbalance with the people.
Here's the thing. The greatest myth of my generation is that votes matter. Votes do not matter. A poignant scene? The Joker's ferry dilemma in The Dark Knight. The civilians voted. What did those votes matter? Votes are a way to excuse yourself from a democracy. They do serve a purpose, and that should be clear when you realize why they don't serve a purpose.
What is a democracy? It is the assumption that people are trustworthy. It is the system of government that assumes that the individual citizen is capable of giving respect to her fellow citizens, is intelligent and knowledgeable enough to comprehend possibilities of "where to go from here", and is compassionate enough to understand and wish the end of the plight of those who are less fortunate.
These assumptions are not true. This is why democracy has such difficulty: because it relies on the respect, intelligence, and empathy of citizens who simply have not been able to give it. There are reasons why they have not been able to, and arguments why they could, now and in the future, but that's about how to fix a democracy, not why we should have it to begin with. So I'll table that. Here's why:
A democracy says that every citizen must contribute to its governance. A government is, by its nature, a subset of the people. In a monarchy, the government is 1/N of the people. In a democracy, the government is N/N. In reality, this is a lot less simple: the monarchy needs a bureaucracy to which it delegates authority, and a democracy's population will never have 100% participation, not least because it's impossible to actually have 300 million individual people in your head. (Through this lens, the big government / small government debate becomes particularly insidious.)
A democracy says that you are a capable governor. That your voice should be at the table when policy discussions are had. If this isn't true, that's a failing of the education system, whose purpose is to provide useful adults who are capable of contributing to those policy discussions intelligently. That's a little chicken and egg, I'll admit, as is the fact that it's an infrastructure failing for you to (1) not be aware of the discussions, (2) capable of being present for the discussions, or (3) too busy making a living to join in.
These arguments shouldn't be a huge surprise. The concept of democracy is not terribly complicated. The difficulty comes in implementation, and in keeping the concept reasonably unpolluted from that implementation. The question of "Why democracy?" comes down to "Can you trust your fellow citizen?" Democracy relies on an affirmative answer. To the extent that you say no, the reasons for the negative are the things that need to be fixed: either the reason is wrong ("he's subhuman nigger; how can you trust him to make a good decision?") or the reason is right ("he doesn't even understand that two is bigger than one; how can you trust him to make a good decision?"). Democracy's job is to transmute "no"s to "yes"s.
Strangely enough, yes: that does sound a lot like getting votes.
> I also think about tragedy of the commons type of problems, which a democracy should have an especially hard time of answering.
To put it shortly, that's what the rule of law is for.