We are told of a wise king who ruled with the Triforce (whole) in the AoL manual. He sealed the piece of Courage in the Great Palace, and Zelda has Wisdom. Since the Prince inherited the Triforce "in part," he got Power.
Bolded is an assumption.
We don't know if that particular prince inherited the Triforce in part for that reason; although with AoL's limited backstory, it would suffice to explain it, and be a simple explanation at that. Daphnes, too, is a wise king who ruled with the Triforce, and seems to have inherited his Triforce fragment (the Triforce "in part"). What becomes of the Triforce after his wish is a big unknown. It could have been hidden; it could have returned to the Sacred Realm.
And TWW.We only have two games in which the Triforce is united, ALttP and OoX.

I place TMC before OoT because it describes a fledgling kingdom and acts as an "origin story" for the Four Sword, Vaati, the hat, the Light Force, etc. Its backstory also does not mention a flood. It mentions wars, which the OoT backstory also references.
Actually, the OoT backstory mentions a "bloody history of greed and hatred", which TP elaborates upon as struggles over the Sacred Realm; not a hero's struggle against monsters.