Besides the overall organization benefit, it would lead to many potentially interesting crucible games, and would make it much easier to switch back and forth between PvP and PvE builds. With 2 or 3 different loadout options, you could preset different subclasses(and abilities), armor, weapons, shaders, and emblems. It'd be much easier than having to switch out all of your weapons and armor every time you want to go between Crucible/Story/patrol/strike. It would aslo be convenient for people who love playing different playstyles. I'm honestly surprised this hasn't been added, or even suggested (At least as far as I know.)
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I think you can on the website (not from mobile)