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.