Instead of looking at the mechanics of multiclassing, 5E's designers should take a step back and look instead at the intent of the player.
Why do players want to multiclass? What roleplaying or mechanical void are they trying to fill? Which desires aren't being met by the current lineup of classes?
I suspect that most players who want to play multiclass characters fall into two camps:
1. Minmaxers looking for optimal mechanical efficiency. I was in this camp in 3E's multiclass-friendly system. I
abused the heck of it, winding up with a rogue/ranger/fighter/wild runner/scout/shadowdancer/dread commando with a flying, incorporeal, can-only-be-harmed-by-magic undead companion. That character was
sick, and could solo most anything. Sigh. Ah, 3E.
2. Roleplayers looking for a little differentiation. A dash of flavor to an otherwise vanilla straight-class character.
What I want the 5E designers to do is ignore the first camp. That game already exists, and it's called 3.x (including Pathfinder).
Instead, focus on the second camp. It's a much bigger camp. And it keeps your game out of the perilous waters of Multiclass Reef, where the edge-case of multiclassing forces undesirable design constraints on base class design: see the previous suggestion to space the poor Fighter's weapon and armor proficiencies over 3 levels. Ridiculous, except if you have to take 3e-style level-dipping into account.
So don't take it into account. Just don't do it, at all.
Focus instead on the desires of the roleplayers. Start by adding cantrips or single armor or weapon proficiency to the existing Backgrounds, as appropriate: cleric cantrips for priest, mage cantrips to sage, longsword or chainmail to Soldier, and so on. Then, give everyone a first-level feat that can only be spent on a restricted feat list: the Disciple feats for those that want to dabble in casting, a suite of similarly-powered feats for dabbling in non-caster classes, and a small collection of low-power generic-but-universally-useful feats (improved initiative and such) for those who don't want to dabble.
Finally, throw a bone to those seeking true hybrids by (shudder) bloating the class list. It's inevitable anyway given the demands of the RPG marketplace. Might as well make class-bloat a feature by introducing, via supplemental rulebooks, new classes that are hybrids of all the basic class combos. ftr/wiz, ftr/rog, ftr/clr, wiz/rog, wiz/clr, clr/rog. Instead of a straight mish-mash of class features from the source classes, these would be all-new classes thoughtfully designed to present a true hybrid experience. No need to hybridize classes that are already hybrids (bard, ranger, paladin). And barbarian, druid, and monk don't feel like they lend themselves to hybridization.
That seems like a straighter, better path. Satisfies the needs of everyone, prioritizing roleplayers first, and not repeating the... mistakes is the wrong word, let's call it simply "choices" of previous editions.