I actually kinda get the question, but I think Link's personality makes it not apply to him. As North Apple says, he's selfless. Think about some of the side quests you have the ability to do in game: help baby chicks grow up. Save cows from 'them'. Help a dance troupe choreograph a 'new' dance. If you've only got three days to save the world, you'd think these would not be urgent tasks, but if you complete the game 100%, you have to take all these actions into account. If he's this selfless about the little details, I don't think leaving a major task like "saving the world" undone is even an option for him.
I also think confidence could have something to do with it. It's just speculation, but Link may have been more assured of his ability to defeat Majora than someone else. To quote Tatl:
...Oh! Hey, you were pretty good out there!
Have you done this before or what?
Link's already saved the world in Ocarina of Time. I don't think he should be overconfident, nor do I mean to imply that it wouldn't be challenging or stressful. No matter what it would be an intense experience. But he's already proven himself once. For a metaphor, I watch tennis. When a players is several points down, forced to play a tiebreak, or something like that, it's a tense moment, but the players who have been in that situation before and gotten out of it in theory should have less doubts about it than in the beginning of their career. It's just psychology.
Of course, I think he believed in himself even in OoT, but having past proof can't hurt. Combine this with his personality, as well as other factors already mentioned, then I can't see him giving up halfway through (as by the time he gets Epona he's already defeated two temples.)