TO answer the actual thread question, "There is no reason to allow Multiclassing [in your game] unless you want to." As the DM, whatever your reasons, if you don't want to in your games, then don't.
While this is true, in terms of GMing best practices, there are some reasons that are better than others. And a GM probably should review their reasons, to be sure they actually achieve the results they want, and those results are also good for the players.
As you astutely note, there is no "character" reason to permit it other than for "min-maxing" or, really, and more accurately, "power-gaming." The only reason players would argue FOR multi-classing is because they want to have their cake and eat it to. They want to cherry-pick their character abilities and they can't do that without an MC level of having access [eventually] to alllll of a class' features.
Gah, no. Sorry, you are, objectively speaking, incorrect. There are reasons to allow it - reasons players can want it that aren't power gaming.
Note, there's a difference between "wanting to pick their abilibties" and "power gaming". Not all desire to pick abilities has to do with increasing power. As has been noted - perhaps the fighter finds (a) god. Picking up some clerical ability is probably not a power-win for a fighter, but it may make sense for the character to do so, in terms of story.
The assertion that there might be "story" reasons to allow it...that's kinda...well, bogus.
No, your assertion that other reasons don't exist is bogus. And kind of insulting. It is effectively a claim to know why everyone else does things, despite their personal assertions to the contrary, which comes across as pretty darned arrogant, too. Just so you know.
 
				 
 
		 
 
		 
 
		 
 
		 Take a look at the charop threads on the WotC board - any weird mechanical loophole being exploited is through multiclassing. I can get x, y, and z by multiclassing with absolutely no drawback where I would have had to spend a feat or made some sort of sacrifice otherwise? Why wouldn't I? And it only gets more and more unbalanced as more material becomes available. It becomes a headache or forces all players to do it to keep up. So I just remove the temptation. I actually feel as though 1e and 2e did multiclassing best - when you multiclassed you picked 2 or 3 classes at character creation and were "all in" for the rest of the character's career - you didn't cherry pick... you were a fighter/cleric or a ranger/magic-user from level 1, just a level or 2 or 3 behind everybody else. I wish they had some optional rules for that in the DMG.
 Take a look at the charop threads on the WotC board - any weird mechanical loophole being exploited is through multiclassing. I can get x, y, and z by multiclassing with absolutely no drawback where I would have had to spend a feat or made some sort of sacrifice otherwise? Why wouldn't I? And it only gets more and more unbalanced as more material becomes available. It becomes a headache or forces all players to do it to keep up. So I just remove the temptation. I actually feel as though 1e and 2e did multiclassing best - when you multiclassed you picked 2 or 3 classes at character creation and were "all in" for the rest of the character's career - you didn't cherry pick... you were a fighter/cleric or a ranger/magic-user from level 1, just a level or 2 or 3 behind everybody else. I wish they had some optional rules for that in the DMG. 
 
		 
 
		 
 
		 
 
		 
 
		 
 
		