Pretty much exactly what you say. In order for the game to be 100% perfectly balanced, we would have to have 100+ champions that have Sona's tankiness, Caitlyn's range, and do 1 damage per auto attack (with no ability to increase damage) and 2 damage per ability, with each ability costing the exact same.
The simple reason behind why we can't achieve balance? Incomparables. Incomparables are a great way to balance a game, but they're also a hard way to balance a game. Incomparables are those skills that would be considered apples to oranges (example being Caitlyn Q (pretty much pure damage) and Sejuani Q (Gap closer/opener + CC)). You can't balance between these two skills, or incomparables, because they are in no way similar. This allows for greater diversity in the game while maintaining the overall balance, pending one of the skills isn't GROSSLY out of balance.
Also, Extra Credits <3
Good Video.