I think that the fact that multiclassing is so widely used is probably a sign that it is working as intended.
I don't think that logic follows. After all, what sees a lot of play is often what is overpowered, and few developers intentionally create things to be overpowered.
IMO, I think all that can be said is that, with enough system mastery, any designed-in drawbacks to the current multiclassing design can more than be compensated for.
And let's not forget that 1e multi classing wasn't well designed either, balanced by race themed level limits.
No, I think that's wrong. The racial level limits were intended to make humans more appealing because Gygax wanted the game to be human-centric. The multiclassing rules
give back to those races, allowing them to do something humans couldn't and allowing them to spend more XP than they would single-classed. Remember, the racial level limits apply to single class characters, too. This means that a single classed elven fighter can only take advantage of 70,000 XP (lvl 7), and an elven magic-user can use up to 375,000 XP, but an elven fighter/magic user can use up to 750,000 XP, even though 300,000 of it literally does nothing at all.
The real trouble with multiclassing in AD&D was because of the XP table. Through name level the amount of XP you need to get to level N usually equals twice the amount of XP you needed to get to level N-1. This meant that while it took 70,000 XP to be an 7th level elven fighter, a fighter/thief with 70,000 total earned XP would be a level 6 fighter. Being single class was just a terrible value proposition in AD&D if you had MC access. The tables weren't 100% lock-step like that, but they were close. The 375,000 XP vs 750,000 XP in the prior paragraph sounds absurdly different, but the 750,000 XP is what makes a fighter 7/magic-user 11
or... a single-classed magic-user 12 or single-classed fighter 11.
The ultimate fix would be making all classes gain subclass features at the same levels (like once in each tier of play for a total of 4 subclass features), so they could create multiclassing subclasses (instead of multiclassing with, say, sorcerer, you take the sorcerer subclass on top of whatever your base class is, at level 1) and be done with multiclassing entirely.
I think that could work, but it would require a vastly different design than the one in place. It would also
only support dipping multiclass.
No to all of this. MC works fine with the exception of warlock. Maybe tune things there and most cheese goes away.
I think you're ignoring the biggest cost of MC in the rules: The fact that classes don't get going until level 3. If there were no multiclassing, then the game would just start at level 3. Level 1 and 2 wouldn't exist. Level 3 would be level 1. You'd maybe even start with 3 HD for survivability, and you'd go from there. That's why character level 1 takes exactly 1 adventuring day to complete, and character level 2 takes 2 adventuring days to complete. Both character levels 1 and 2 are meant to be over with
immediately. Class levels 1 and 2 are designed to be a multiclassing tar pit. This is all so that the game can let you do a la carte multiclassing, and punish you by making it cost 2 null levels to do it. The problem with 5e is that some classes (Warlock, Paladin) are still too front-loaded. Warlock is even worse because the class feature table feels empty from level 4 to about level 11.
Personally, I think I would be happier with multiclassing if there were benefits to keeping the classes within 1 level of each other. I'd even be a fan of making all the classes 10-12 levels long and then requiring multiclassing to progress past level 10-12. Given that most features you gain at those levels seem to be either wildly useful or completely pointless or else totally broken (e.g., 7th-9th level spells). With each class capped, the optional multiclassing rule becomes whether you require players to complete the first class before beginning the second, or else if you allow players to start swapping back and forth immediately. Unfortunately, I don't think it would be accepted as D&D.