Who says that a game of chess must be played with the king's conventional movement restrictions? If you want to use a variant rule that makes the king more powerful, then find an opponent who would be willing to play a variant of chess.
For a long list of interesting variants for chess, search for "chess variant" on Wikipedia.org (or just follow the link below).
http://en.wikipedia.org/wiki/Chess_variant
In an endgame with little material left on the board, I don't think many people would argue that the king is often the strongest piece, even if it can only move one space.
If you would like a practical demonstration as to why the king should not move more than one space, please come over and play against my 4 year old son (or go find a 4 year old of your own). He likes to move the king (and the pawns) as many spaces and in whatever direction he feels like, and while it is very entertaining to watch, it is not enjoyable to play, even if you follow his rules.