IMO Sakurai was following the model of Joseph Campbell's "Hero's Journey" when creating the story of Smash Bros.
This fits in with Master Hand being a "final boss". Since the fighters are trophies and therefore toys, the Master Hand is the hand that plays with those toys. This elevates the hand to the status of "Father" (Not a literal father, but in the context of the Hero's Journey the "father" figure is a symbol of (possibly absolute) power and authority). In this sense, the fight with Master Hand is the "Atonement with the Father" step of the Hero's Journey, which is usually supposed to be the peak or climax of the story. This is backed up by the fact that, once the Master Hand is defeated, the fighter turns back into a trophy, therefore completing the "Apotheosis" (or transformation via death or a death-like state) step of the hero's journey. Shortly after that you gain "The Ultimate Boon", which in this case is the character's trophy, which is why you went through Classic mode to begin with.
Not strictly an unpopular opinion, since many don't know what "The Hero's Journey" really is, but it's how I choose to interpret Smash's story line: As a loosely-followed formula instead of a story.