Simple election methods break when more than two candidates are available.
Do they? Or are the voters just failing to vote effectively?
For example, imagine that a significant minority cast Reform party ballots in the upcoming election. Most of those conservative voters probably voted Republican in previous elections, so this would mean a decreased Republican vote. The only practical effect is
to weaken the Republican candidates against their only real rivals, the Democrats. (Similarly, a strong minority turnout for a third, liberal party would in practice serve only
to weaken the Democrats.)
I think that Minnesota Democrats and British Conservatives wish that were true, but it's not. Democrats lost against a strong Reform candidate for Minnesota Governor, and Conservatives lost their majority in the British Parliament even with a significant minority of Democrats around.
Therefore most voters feel that alternative candidates not only waste votes, but may even strengthen the major party farthest from them.
This isn't a flaw in the system, but rather a flaw in the reasoning of the voters. There exist third parties representing all sorts of views, not just one. And some parties won't draw votes from just one major party. From whom should the Libertarians take votes?
Add to this the electoral college (a bizarre historical complication to U.S. presidential elections), and we end up with an system that fails to adapt to or accurately reflect
Maybe you should explain why you call the electoral college "bizarre," and why you think the system fails the people.
Look up the 1860 Presidential election. Was that a failure?