I don't know about the USA, but here as soon as a woman or child is murdered they immediately become "Tragic Sarah" (or whatever) in all the headlines. This is a journalistic convention (and one which smacks of laziness) and has nothing to do with the sense in which the word "tragedy" is used wrt literature.

A tragedy is when a hero brings about their own downfall due to a flaw in their very nature which they are simply incapable of overcoming. Humanity bringing about its own downfall by continually going to war fits the bill perfectly. In a Good vs Evil story such as LOTR, neither side brings about its own downfall – it's the enemy that does so. The closest thing LOTR has to tragedy is the story of Gollum (which I've always thought was Tolkien's masterstroke).