One of the goofy things about the US is that we so much take it for granted that democracy is the best form of government -- and all other forms are in turn stigmatized -- that we really don't think about it very much or very carefully, which I think winds up producing a fair amount of sloppy thinking when we get around to doing institutional reform. I'm not going to go through all of the ideas behind democracy in this post, but I'll just say that most of us have been mainly exposed to Lockean, liberal reasons for democracy, which I think for most people come down to questions about outcomes: democracy is best because it produces, or at least promises to produce, the best, or fairest, or most justifiable public policies. Yet there are also participatory arguments in favor of democracy, which are related to republican arguments generally associated with a long line of thinkers often traced to Machiavelli. Theorists in that tradition argue that public action is, at least for some people, self-fulfilling; that is, they believe that one of the things that set humans apart is that we have a capacity for collectively organizing the way we live, and those who get involved in politics for whatever motives often find that it is deeply satisfying. Hannah Arendt talks about this capacity for "public happiness" extensively in On Revolution, noting that people caught up in revolutions often express surprise that getting involved was so personally fulfilling; see also Gordon Wood's discussion of the American revolutionaries, or, for that matter, what veterans of the civil rights movement have said about it. Arendt makes the point that Jefferson's odd formulation of "life, liberty, and the pursuit of happiness" contains an ambiguity (one can see our discomfort with that phrasing by watching the excellent Schoolhouse Rock on the Declaration -- if you haven't watched it recently, check out how they deal with pursuit of happiness). Read as private happiness, Jefferson's formula is a good liberal equivalent of the more common "life, liberty, and property." By that reading, government's main function is to protect individuals from encroachments on their private affairs, and at the same time to support those private activities, even if it takes government action to do so. But read as public happiness, Jefferson is asserting that the opportunity to participate in collective self-government -- something that can only happen in some form of democratic republic -- is a core human right.
Which gets us to the 1780s. As Gordon Wood writes (see generally Part Four, especially Chapter X section 5), what alarmed the revolutionaries was that their assumption that an American republic would deliver both types of happiness was undermined by a turn away from public involvement: on the one hand, representative republican legislatures appeared shockingly prone to tyranny, and on the other hand the people seemed content to ignore politics entirely as long as they were left alone to enrich themselves. Self-interest, in other words, appeared to be everywhere, and most shockingly it appeared to be perhaps the natural result of the revolution, thus making its higher goals self-negating. Remove tyranny and establish republican institutions, and people will -- now that the threat of tyranny is gone -- turn away from public affairs to self-interest, thus establishing the conditions for a new, democratic version of tyranny.
In my reading of Madison -- which I should say from the start is contentious, and hardly the consensus view -- the Constitution is a brilliant attempt to escape from that trap. Centuries of Whig and republican thought had assumed that their project could only succeed if the people were virtuous, and public spirited. Madison says: what if we turn the tables on all of that? What if we take self-interest as a virtue, or at any rate as inevitable in a democracy, but use it against itself? The genius of Madison's Constitution is that it encourages political participation even if it is originally motivated by selfish gain or ambition, but then counts on the complexity of the system to force people to actively engage in politics if they hope to get anything done. Simply registering one's preference or making demands will never be enough. Moreover, by separating institutions through checks and balances and federalism, not only is tyranny avoided in the sense that private citizens are protected, (good for private happiness!), but also in the sense that opportunities multiply for citizens to become meaningfully involved in public affairs (good for public happiness!).
Of course, if you don't believe that participation in politics can be good for its own sake, then you aren't very likely to care very much about that latter virtue of Madisonian democracy. Even if you are, you may still believe that the costs of Madisonian democracy, the difficulties in translating popular preferences into government policy, aren't worth the benefits (although as usual, I'd caution people about assuming that election results can be easily translated into preferences for specific public policy choices). And even if neither of those things bother you, if it turns out that capacity for enjoying and thriving at politics is unevenly distributed (the way that capacity for self-fulfillment in, say, music or fine art appear to be unevenly distributed), then it's not clear how we can justify the unequal influence over government that would result from natural sorting. In other words, I don't really have any answers at the end of this long post. What I think I can say is that those who write about public happiness, either in their personal stories or as political theorists, tend to emphasize the notion of discovery: people who became involved in politics for other reasons, whether it was self-interest or to right what they saw as an injustice, were surprised at what they found -- you hear versions of the idea that they felt truly alive for the first time, or something to that effect. One could even perhaps bring in Ronald Reagan's notion of finding "the rest of me" in politics, no? All of which suggests to me that one should not necessarily count the apathy of the uninvolved to be conclusive. I'll end with a quote from Arendt (On Revolution, Penguin ed., 131):
And Jefferson's true notion of happiness comes out very clearly...when he lets himself go in a mood of playful and sovereign irony and concludes one of his letters to Adams as follows: "May we meet there again, in Congress, with our antient Colleagues, and receive with them the seal of approbation 'Well done, good and faithful servents.'" Here, behind the irony, we have the candid admission that life in Congress, the joys of discourse of legislation, of transacting business, of persuading and being persuaded, were to Jefferson no less conclusively a foretaste of eternal bliss to come than the delights of contemplation had been for medieval piety.