Why does the game end at checkmate and not upon taking the king? Ever since I was a kid this has felt kind of odd to me. Victory in chess is achieved by "cornering" the king, and not by taking him outright, like what you do to all the other pieces. I always wondered why that is. 

Of course the distinction between taking the king and "cornering" him is very minute, and upon achieving checkmate you can confidently assume that a capture will occur next turn, but somehow it feels odd that this next step is forbidden by the official rules. What if the player who achieved check-mate is inexperienced or distracted and doesn't realize that he did so? What if he's really experienced and wants to continue taking material for sadistic purposes? Why do the rules of chess disallow such moments? These moments probably wouldn't end up happening in tournament games (since most seem to end up in resignation anyway), but nevertheless they could, and I think chess would be more fun like that. So why was the checkmate created?
Per wikipedia, citing Henry Davidson's A Short History of Chess (1949):


  In early Sanskrit chess (c. 500–700) the king could be captured and
  this ended the game. The Persians (c. 700–800) introduced the idea of
  warning that the king was under attack (announcing check in modern
  terminology). This was done to avoid the early and accidental end of a
  game. Later the Persians added the additional rule that a king could
  not be moved into check or left in check. As a result, the king could
  not be captured, and checkmate was the only decisive way of ending a
  game.