Why does losing lead to poor self-esteem? Well, perhaps it is because most competitors lose most of the time!
Think about it statistically. The best probability is a 50/50 chance when the competition consists of two teams (or just competitors). So when competitions increase to more than just two competitors, the odds change and each team has a better chance to lose than to win; that probability of losing directly increases as the number of competitors increases. As such, people who compete are exposed to much more failure than success, whether it be athletics or a simple contest.
Further, this is just a macro view of failure being rampant in competition. Specific examples such as the science (or art) of hitting in baseball drive the point home. The best hitters in baseball, on average, get a hit three out of every ten times they come up to bat. That means that the greats who are enshrined in the Hall of Fame failed seven out of every ten times!
If people feel the need to prove themselves worthy by winning, they're in for a rude awakening. The worst thing we can do is attach our self-esteem to whether we win or lose at any competition.