(This is actually true of all voting systems, not just the US Government.)
It seems like common sense that the purpose of voting is to determine the winner. But like much common sense, it is wrong.
The purpose of voting is to satisfy the loser that they lost. Determining a winner is easy, it is the guy with the most guys and guns who is still alive. A voting-based system depends on the acquiescence of the losers, so that they don't just grab guys and guns and take over.
To the ears of someone raised in the United States, that sounds weird, but it happens all the time in the real world. And there is certainly no natural law that says it can't happen here.
There are two parts to such a system: Participants who agree to live by the results of the system, regardless of results, and a system that satisfies them all. Technically, this is all that is required; other things like "fairness" are not.
As long as all participants agree in advance that a vote for A is worth two votes for B, it is OK. This isn't merely academic; we use this standard all the time. One example is in passing a new amendment to the US Constitution: As a first step "two thirds of both Houses" must pass it. Requiring a "two thirds" vote is equivalent to saying a vote against is twice as strong as a vote for. Phrase it as "requiring 2/3rds", most people would agree that was fair if it was agreed to. Phrase it in terms of differential vote strengths and I bet people would call it unfair, even if agreed to in advance. But "fairness" is not necessary, only agreement.
Clearly, the 2000 Presidential Election in Florida failed to meet this standard for a successful election, as the loser did not gracefully acquiesce soon enough. As a result, confidence in the system was degraded. However, the failure here is not really the candidate himself, as both Gore and Bush lived by the results of the other 49 states without (much) contention. The failure rested more in the second part, the voting system, as a combination of various perceived flaws in the system that were not part of the implied agreement, and the fact that our vote counting system is not infinitely accurate, combined to produce a result that was not mutually agreed upon.
It is not possible with a paper voting system and the set of humanity that currently votes to avoid marginal votes. That's the reason I consider it a real pity that we have so royally screwed up electronic voting; it had the real potential to make voting much more accurate and make the results much more reliable, instead of much less reliable in the absence of a paper trail.
In fact, this myth explains why it is so hard to convince people of the importance of a paper trail. We don't need to convince the winner they won, we need to convince the loser they lost. And speaking as a guy with a Master's degree in computer science, years of experience, yadda yadda yadda, if I were in an election, mere numbers on a screen could never convince me I lost. (I try to be fair, so I wouldn't be convinced I won, either, but most winners would not be so gracious.) Only the paper trail can convince the loser they lost.
Elections make a lot more sense from this viewpoint, as do the failures of elections, and the dangers of toying with elections. I am very afraid that lawyers think they can continue to toy with elections because the system won't fail until it fails to register a winner. Since we are currently quite far from that, they may think they have a lot of room to play. The truth is that elections will fail when the system fails to produce happy losers, and we are far, far more close to that, with far, far less room to play, than I am comfortable with.
(Vague hat tip: I'd swear I saw the kernel of this idea at Den Beste's site, but if so I can't find it. I know the kernel of my coalition ideas in myths 1 and 2 came from this post and the links to his other essays in that piece. Update: I think I found it here; the ideas are there but I am bringing them out in this series with a different focus.)
(This is #3 in the government myth series.)