The reason leaders don't fight is because the first nation to not fight by these rules will win, and whoever doesn't form an army will lose their stuff.
Angela Merkel decides that she's the rightful owner of Tokyo and declares war on Japan. In response, Shinzo Abe decides to step down and let Kisenosato Yutaka take control of Japan during the War of Merkel Aggression. See the problem? Countries would change their laws regarding leadership in order to make a good fighter the leader until the war is over. World 'leaders' would just be figureheads to fight the wars while the actual leaders control the country, much like armies and leaders today. However, a neat side effect of this system would of course be the rise of the Icelandic Empire.
Even if one a leader wins the war, what then? Will everyone in the country say "fair fight, it's your country now, we'll leave"? Of course not, they'll take up arms and fight the leader. And since the leader was just beaten, the people in the other country will take up arms to defend their stuff, and suddenly you're right back to two armies fighting.