If two teams play a tremendous, very even game to the end of regulation, why should one go home with a loss due to a ball that just nicks the foul pole in the 13th inning or because it lost the coin toss or because a field goal attempt hit the upright?
Personally, I advocate the elimination of all tie-breaking mechanisms (overtime, shoot-outs, sudden death, whatever you call the thing that college and high school football do) in regular-season contests. If teams are tied at the end of the allotted time (or number of innings), then neither deserves to win or to lose.
Of course, no one would ever go for eliminating all tie-breakers, so why don't we just get rid of tie-breakers that involve changing the way the game is played. For example, the NHL's shoot-out is a fundamentally different type of competition than the game they play during the 60 minutes of regulation. College football's "overtime" removes football essentials such as managing field position and the clock. Let's also eliminate overtime formats in which the result of a coin toss gives one team a significant advantage.
But even if we eliminate these tie-breakers during the regular season, we still have the problem of breaking ties in post-season tournaments. In these situations, I would advocate simply continuing regular play, as in basketball overtime or extra innings in baseball. For football, this would involve two extra periods, each beginning with a kick-off and ending when no time remains on the clock. Eliminating nonsensical tie-breakers could result in some very lengthy post-season contests (especially in hockey and soccer), but maintaining the integrity of sports requires sacrifice.