Anachronistic: the electoral college arose on account of a number of factors. Consider that the biggest challenger was Congress electing the President, which faces obvious flaws. One of the advantages suggested for the electoral college system was that the single-purpose entity would be less prone to the politics and intrigue of a standing body. Hamilton also argued that the distributed nature of the electoral college would insulate the decision-making process from potential outside influence. It was also seen as an advantage to build a firewall from one state to the next, so that issues in one state would not unduly influence the election. But there was also a measure of elitism to the electoral college, the idea that the political class had more information and greater faculties with which to make the difficult decision.
It's useful to note that Hamilton and Madison argued forcefully against states binding their electors to a single candidate. The assumption previously was that electors would be a respected local figure who could then deliberate thoughtfully, rather than being bound to a particular political party. Maine and Nebraska are the only states still living up to the framers' original intent.
Hair-brained: the previous discussion of the unexpected political effects of the electoral college should be enough to prove this point.
Undemocratic: this is a tough one, since as you pointed out, "democracy isn't some magical pixie dust." However, I think it's fair (at least when discussing the American political system), to situate our various institutions along an axis of democratic to republican. To the democratic end of the spectrum are institutions that provide a direct role for citizens in politics, such as caucuses, referenda, and (ostensibly) the presidency. To the republican end are the representative institutions. Given that the presidency was intended as and is best seen as the most democratic federal institution, it is unfortunate that the manner of election still has this one wrench, where the vast majority of citizens have zero practical impact on the result.
