The whole impetus behind 5e was to have a game that everyone playing at the same table could enjoy as their favorite version of the game, by allowing each player to bring their favorite edition's style into the game, at least from their perspective - to have players that preferred Basic at the same table as AD&D players, 3e players, and 4e players.
That was an early idea put out there, but they really seemed to back off from it as Next developed.
IMHO, 5e has mostly succeeded in allowing that to happen, by bringing very balanced options to the table with their modular approach.
Couldn't disagree more. 5e brings excellent support for the 2e fan, a fair bit of related nostalgia for fans of yet earlier editions, and, with Feats & Multi-classing opted-in, is not terribly disappointing to the 3.x fans (though, really, offers little reason to switch back from Pathfinder if they'd already gone there). But, it doesn't quite deliver on allowing 3.x and AD&D fans to play at the same table, each feeling like they're playing their favorite system. And, to try to get a 4e experience out of it, you have to have the DM opt-in to modular options that are incompatible with those AD&D/3e experiences at the same table, and, even then, fall far short of delivering on the 4e-style experience.
I don't blame WotC for that 'failure,' though: the idea in question was overly ambitious, and they never committed to it formally. 5e succeeds admirably at bringing back a classic D&D feel, with some of that 3.x character customizablility and d20-compatible mechanics, while delivering on DM empowerment in spades. That's still a damned impressive achievement.
If you can not see the benefit or ability to play an effective, interesting, and well-developed character in D&D without multiclassing, you are precisely the kind of person that needs to sit at a table and play without it.
3+e style MCing has been the source of a [often unrecognized] tragic degradation of the role-playing game...."my opinion", of course.
It will NEVER have a place at a table I run.
Sounds like you're precisely the kind of person who needs to sit down at the table with some cool MC'd characters. ;P
Seriously, though, I couldn't easily disagree more with your characterization of 3.x-style MCing. It was probably the single best idea in 3e. Had the class designs been worthy of it, it'd've been a remarkably elegant and efficient way to enable a virtually limitless range of character concepts, with only a relatively few & simple character classes.
Unfortunately, that kind of 'modular' multi-classing demands classes that are so neatly designed and robustly balanced that taking the 1st level of any class is as viable as taking the next level of any other class, at any level. And D&D's never come close to that. The 3.x fighter could have been the model for designing classes that'd've really worked with that kind of MC'ing, but no other class was ever implemented in a similar way. 3.x also screwed up save advancement and caster levels for it's own MC system. 5e, for it's part, fixes the latter, AFAICT.
In any case, there's nothing about being able to do a better build-to-concept that gets in the way of RP, rather, it supports RP by giving you something closer to the character you want to play. Sure, customization options can be abused to break the game or just used badly, but that's true of any sub-system that has any value at all.