You hit it spot on there. Only for me, it's less of a psychological aspect and more of a character development one.
Sure, Twilight Princess did have a darker story than most Zelda games, and the ending did have some bittersweet elements, and the darker-colored graphics certainly give it a melancholy feel. I will grant TP that.
However, Majora's Mask had something that no other Zelda game has been able to recreate: an emphasis on characters. These weren't characters you were trying to save; they were actual people. They had schedules and routines. They had desires and ambitions. Almost every character had a personality and some sort of character development. These were real people you were trying to save. And to top it off, as many people as you can help, there were others (like the swordsman--you can save his life, but he'll still be a vain coward) whose problems went unresolved, and every time you reset time, their problems would become unresolved again. Also, notably, the game places a large emphasis on the number four. In Japan, "four" and "death" are homophones.
Ironically, even though it's (in my opinion) the darkest game, it's also the one that contains the most hope. While in other games there is little character development, Majora's Mask gives you the chance to actually help people with their problems. You heal the troubled souls of the deceased. You allow a troubled man to vent. You help a depressed Cucco farmer get his final wish. You clear misunderstandings and reunite Dekus with Monkeys. And quite possibly most importantly, you save a child from being abducted by aliens and keep a marraige from falling apart. The best part of all this is that, if you play the inverted song of time and use owl statues, Nintendo made it possible for you to accomplish all that needs to be done in one day, right before beating Majora. Sure, some things conflict, but the Cuccos will grow up on their own, and there will be more shipments of bomb bags. As dark as Majora's Mask is, it is also the most inspiring. Which is why it will always be my favorite of the games.
And on an unrelated note, this topic should really have a poll.