"Democracy" and "freedom" are wholly compatible with promoting economic equality, given the understanding that "freedom" means the right of the people to exercise their will through democratic institutions.[1] If promotion of economic equality is what benefits the greatest number, then it is natural to assume that free people will exercise their will to minimize economic inequality (as they have in social democracies in Europe).
[1] To a leftist, and indeed many other people, what separates authoritarianism and free society is not the lack of someone telling you what to do, but who that someone happens to be. Arguably, a classic libertarian government is indeed authoritarian: state power is exercised pursuant to the policies of philosopher-kings who believe in a particular set of "rights" (private property, etc), not pursuant to the will of the people themselves (who may very well have different beliefs). Indeed, even as someone who isn't a leftist, I have a hard time seeing much difference between classical libertarianism and divine monarchy. Whenever I hear talk of "natural rights" I think about who kings once justified their unilateral exercise of state power by reference to divine natural law.
The obvious difference between a libertarian government and divine monarchy is in the level of authority given to the government. In a monarchy, the monarch's will is law. Under libertarian government, the government is extremely limited to only protecting a very small set of rights. You could in theory have a libertarian monarchy where a king was only given a limited authority to enforce laws related to natural rights violations, but the two concepts are very different.
A libertarian democracy of course is more desirable. Which means the people's votes, or the authority given to their representatives, would be limited to only those issues related to natural rights violations. Under a libertarian government, the people are still quite free to exercise their will, only they could not use the government to impose their will on other individuals. Social problems can still be solved through community organizations outside of the government.
I think the biggest weakness of the pure libertarian philosophy is the uncompromising defense of property rights. If a small group manages to acquire all of the property, they can use their monopoly to exploit everyone else while the government protects their "rights".
My view is this: once you bind everyone under the monopoly on violence that is government, its wrong to not let the people who are bound decide what the government should do. That's the fatal flaw of non-anarchist libertarianism to me. Its some cabal getting together to decide what "natural rights" should be then using the monopoly of violence to protect what may be their minority viewpoint. Appealing to "natural rights" is no better than appealing to "divine law." Its self-serving hand waving.
That's the old "there is no morality without God" argument. Just because rights aren't handed down on stone tablets (i.e. natural rights) doesn't mean that there are no rights. In the English tradition, rights arise from the long-standing practices and beliefs of a society. Thus, unless we have a society that embraces savagery and murder in general, it's not "just fine" for the majority to cause the genocide of a minority.
Also, what's worse: a system in which the rights of minorities may occasionally be trampled, or a system in which a minority of philosopher kings by design circumscribes the political self-determination of the majority?
Classical libertarians are in the position of arguing that self-determination (the right of a free people to structure their society as they wish) is slavery, and that the object of peoples' political freedom should be circumscribed to a narrow domain handed down by an Inner Party of philosopher kings.
[1] To a leftist, and indeed many other people, what separates authoritarianism and free society is not the lack of someone telling you what to do, but who that someone happens to be. Arguably, a classic libertarian government is indeed authoritarian: state power is exercised pursuant to the policies of philosopher-kings who believe in a particular set of "rights" (private property, etc), not pursuant to the will of the people themselves (who may very well have different beliefs). Indeed, even as someone who isn't a leftist, I have a hard time seeing much difference between classical libertarianism and divine monarchy. Whenever I hear talk of "natural rights" I think about who kings once justified their unilateral exercise of state power by reference to divine natural law.