I'll admit to being an adolescent when the game first came out... but I replay it regularly and age and wisdom haven't caused me to notice any new problems* outside of the jaggies/blurry textures/framerate issues inherent in revisiting any N64 title. Heck, I actually loaded up the Virtual Console version last night and I've been having a blast playing about 3 hours of it so far, barely even setting foot outside of town. It really is an incredibly rich and unique game where you come to appreciate how much almost pointless level of detail is packed into it, how the sidequests are intertwined, how you can discover a sidequest or story element in a bunch of different ways (which really makes it feel like a 3 day span of time centering around a pair of important impending events), the uniqueness of the three different races and how they all have unique powers, the surprisingly frank and mature stories of these people, the heartwarming way that people express their gratitude as their wants and regrets are eased as the end draws near, the heartbreaking way that it sometimes feels like Link is honestly exploiting them with his powers to get what he wants, the fact that you honestly cannot save and help everyone (something the game makes tragically clear in the ending, even if you 100% it), and even at the very end it is finding ways to be surreal.
People get emotional about Majora's Mask because Majora's Mask is a very, very emotional game that deliberately draws on feelings of helplessness, loneliness, loss, regret, love, redemption, and healing.
*That's not quite true. The one major problem I have now is that I have the sidequests so ingrained in my brain now that I can't really organically stumble across them and try to figure them out any more. I can (and have) do like 6-7 of them in the first cycle after you become human again. I honestly wish I could forget all of that, because learning all of that in the first place is such a wonderful experience.