The fact that the public thinks in a binary matter isn't some flaw in their reasoning or evidence of their stupidity, but a product of the electoral system within which they function. By giving your vote to an independent you especially you prefer, you are taking your vote away from a party candidate which you do not especially prefer, but at least preferred to the other party candidate. This is an illogical action: the independent is unlikely to win, given the status quo, and as such you just make it more likely that your 'least-preferred' candidate is elected. To fix this problem, people would have to be relatively certain that if they started voting for independent or minor parties, everyone else would, all at the same time: a collective action problem of prohibitive magnitude. This is why two-party systems tend to be entrenched in democracies that use single-majority-vote systems. At best you can hope for regional variations (i.e., the two candidates in one area are different to those in another, hence why the United Kingdom sustains the Liberal Democrats), but that's a poor solution. It's particularly poor in America, given that while the parliamentary nature of, say, the United Kingdom's executive means that you can still extend some executive power to those outside the main two parties (hence the current coalition), America's presidential system means the executive's power is placed entirely with the winner of a simple majoritarian system, which has little prospect other than to alternate between the same two blocs. If you want to see where the blame for America's political ills lie, before you look to disengagement, lobbying, corporate influence, stupidity, or any other factor; look to the constitution. You will find your problems begin there.