Multiclassing is an optional rule in 5.0, and I'm glad of it, so I don't have to feel like turning it off is taking something away from the PCs that they were supposed to have.

Anyway here's my list of reasons. Maybe there are more, I don't know.

1- Multiclassing pushes players to separate lore and mechanics.
Warlock works pretty good with paladin, and also with sorcerer. If a player were to pick any of the combinations that seem to work contrary to the intended story of the classes, it begs the question, why even have the lore and in-universe power sources at all?
2- Multiclassing is easy to screw up.
Pretend the players handbook had five hundred classes, and almost all of them sounded awesome, but three hundred and fifty were mechanically very poor. Seems like a lot of work right?
3- Even the good multiclass combinations end up with target levels.
A "target level" is a level where the "build comes online" and whatever synergy is intended occurs. Before and after these levels, the combination often lags.
4- Many classes have strong early features which multiclassing brings out in unexpected ways.
If multiclassing was done correctly, I would expect to see people say "fighter 8/rogue 12 or fighter 10/rogue 10, which has the better progression from 1 to 20?". For choices that actually do evenly split levels and concept, see point (2)- precious few work great.
5- Some multiclass combinations are just too good overall.
I am pretty sure some of them are just more than the game was ever designed around.
6- It's totally backwards to what I want to do.
It wouldn't be an achievable project to have a bunch of partially customized classes, feats, and spells if it was supposed to be for every combination. I'm not sure if what I'm doing, even with its much smaller scope, is really achievable or smart, but I believe it can be. With multiclassing, I don't have that faith.
7- System knowledge tests are bad.
I think anyone can read the documents I linked and pick a thing, and I hope that thing works. If multiclassing were allowed, well, you'd have to study that, right? There were a lot of cool feat combinations in Pathfinder that no one discussed or discovered and that was a game where we had years and a million feats. I think all the information in one place is good. If the next thing I say is "...and you can do everything in every combination", like, who does that help? We don't even have a power gamer to please!
8- It's harder to make a custom thing.
I want to make custom things, and multiclassing makes it hard.
9- Weakens the core concept of a class system.
If every unarmored caster starts as an artificer or cleric so that they can get heavy armor without losing their spell progression, what is going on here? If the classes do what they say on the box, isn't that better design?

Oh one more:
10- Makes it hard to adjust on the fly
If we have someone who goes straight $CLASS and it sucks a lot, say past I dunno 10th level, I believe the right fix is to open my document and add crap to that class chart, not to respec the character so he has more spellcaster levels or whatever. If a multiclass guy were in that spot that seems much harder to do right.

I don't know if "class balance" has ever mattered in D&D, at all. But the game has spilled so much blood trying to make it happen, and it seems like it's close and doable. Multiclassing only hurts that.
There's 15 classes in my Classes N1.odt. I feel I'd rather make 105 classes, one for each dual class combination, and try my hand at balancing 105 classes no one would ever actually play, before I turned on the piecemeal system 5.0 has. I just feel it's a bad system, I can't shake the feeling.