Up until the box art part of me really thought he was going to, but nope. It's an interesting choice to have some of the good guys wearing them at all given that this particular game relies so heavily on the coats as a visual signifier of a characters allegiance/loyalties. I mean if Riku and Mickey's new threads can shield them from the effects of the darkness, couldn't they do the same for everyone else? Perhaps it's a matter of growing pains. I imagine that Roxas, Xion, and Lea in particular (mainly because there's an implied duality of some kind between Lea & Axel in terms of who he sees himself as) are probably going to be struggling with their identities in some form. Two of them's entire lives were spent in service to the organization after all, so it's reasonable that there may be some sentimentality attached to those coats.
Also worth mentioning that given what we've seen in X back cover, the coat's roots appear to stem from that of Ancient Keyblade wielders, specifically the MoM and Luxu, and may have been made for similar purposes to that of keyblade armor (meaning then it wouldn't necessarily be that out of the ordinary) though that's just supposition. Perhaps it could prompt some kind of interesting conversation or tidbit regarding the keyblade's history at some point in the game (though I'm doubtful about that). Idk, something to think about.