Why only keep one exotic? You'd have to switch it out constantly. And yes you can store on your characters as well. I used to run raids and my weekly's with full characters so all drops would go to my postmaster and I used that as storage as well but that is very limited so it needs frequent cleaning.
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Switchcing them out isn't a big deal really, especially with the app. You could also just divide your weapons so you don't have to switch for variety.
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I don't care about variety, I care about efficiency. As the app stands now, switching one or two weapons isn't a big deal. But an entire loadout from one character, to the vault, then to another character; is very inefficient.