I' m very aware of that quote. And he would know how?
The fact is, the game contradicts itself on this matter. Remember that the King was looking for the Hero of Time specifically. It's like Zuko looking for the Avatar. Only in this case, there's genuinely no way he's going to find him.
Many of the games attempt to attach Link to a bloodline of some sort. For the most obvious example, there's OOT Link and TP Link.
Your idea might explain a few things about Link's Awakening. I always figured that in hindsight, these characters were echoes of their OOT counterparts, and what...