But, why do all classes in a RPG be able to do everything as well as all the others?
This is not what balance means. Balance does not mean that everyone is the same. There can be wild differences between classes, that are well balanced.
Ultimately, the important thing is that there be balance between
players. And that can be summed up as: each player should have equal right to claim that they are playing the protagonist of the story.
In narrative games, this is enforced directly, by providing balanced mechanics for distributing narrative control. But in games like DnD, the mechanics are highly oriented towards how effective a character is within the world, and that effectiveness is the prime determination of which characters can truly claim to be "protagonists".
For example, when the casters massively overshadow the non-casters in 3.X, it becomes laughable to think that the non-casters are in any way the equals of the casters. The casters are clearly the stars of the show. Which ends up meaning that the players of those characters aren't being treated as equals. And people tend to not have as much fun if they are being treated as less than the equals of their peers.
In addition, balance doesn't just have to mean combat, but if that is the mechanical focus of the game (as has traditionally been the case in DnD), then those mechanics need to be balanced. In a game with a more even mechanical focus on combat, and non-combat, it's OK for characters to be balanced overall, not just in combat.