I know it is "explained" in the instruction manual, but it is hardly explained. Essentially, this is the "lengend of Zelda" itself, this is the reason all girls born in the royal family are named Zelda (confusing no doubt) and all it gets in the series is a few pages in an instruction manual.

Why not make a game where you play the prince for example. You could start off wanting to be a hero, going around doing good things. Then meet the evil wizard, who you think is good. He fools you into thinking you have to find out about the triforce in order to defeat a great evil, and since you sister wont tell you the secrets of the triforce, you allow the wizard to use a harmless spell to get it out of her. Then after the spell is cast on her and you realize the wizard only wanted your help to gain control of the triforce yourself, you embark on the games main quest, to find a way to awaken your sister by breaking the spell.

Usually Zelda games have a beginning quest where you have to collect a number of item to get to the main quest (Link to the Past, the pendants; Ocarina of Time, the stones from the three temples) after which some events take place upping the stakes and starting you on a new much larger quest (A Link to the Past, Zelda is taken and the portal to the darkworld is opened; Ocarina of Time, you enter the temple of time as a kid and wake up as an adult). In this one, this game the part where you meet the wizard would be in your first small quest, then after Zelda fall under the spell the larger quest would begin.

Something like this would work, and would help answer some of those untouched questions, or at least elaborate on lore that has only been backstory until now (like how Ocarina of Time was essentially the backstory we read all those years before in A Link to the Past's instruction manual and cutscenes. It was really cool to see what was only backstory before unfold in front of your eyes with you as the player).