Except, they kind of didn't. If I played a fighter/mage/cleric in AD&D, playing an EK 7/Tempest Cleric 6/Wizard 7 in 5E is not at all the same thing*
EK Acolyte could cover it, vaguely, the cleric levels possible in 1e were pretty low. Or EK/Paladin, perhaps?
You get cumulative HP instead of averaged HP, your spells are all wonky and you have spell slots way more powerful than a 7th level wizard should have, proficiency modifier and cantrips are too powerful, your saves are wrong (instead of the best of all three 6th-7th level classes' saves, you get only one, but that one class's saves are boosted to 20th level) and your advancement is extremely slow. Not at all the same experience.
Just because two things are both labelled "multiclassing" doesn't mean you can expect them to behave the same way.
So, 5e uses a version of MCing introduced by 3.0, which really is a very intuitive, even elegant system, in it's basic concept. Each level is like a building-block, you stack them up to match your character concept. Less complicated/granular than a point-build system like Hero or GURPGs, still retains the D&D mystique of classes & levels, while providing much-expanded player options without equally-expanded bloat.
All it needed was a set of classes that were equally elegantly-designed to mesh with it.
It never got them.
The closest, by far, was the 3.x fighter (after first level: BAB, saves, bonus feat at each even number level, that's it - aside from which feats can be taken and a very few fighter-only feats), and it was all alone in that regard.
Also completely un-necessarily screwing up 3.x MCing was the way saving throws scaled, and the way caster levels were handled. Y'see, 3.x had 'BAB' instead of THAC0, it was just a bonus that progressed at different rates - 1:1, 3:4, or 1:2 - for different classes, when MCing, you just added them up. (Adding levels of all classes with the same BAB progression together beforce calculating and adding to those of different progressions would have been even better). Saving throws also added, but the problem was more pronounced because 'good' saves started at +2, so take several classes with a good save and you had a huge bonus, while your other saves stayed 0 (again, obvious solution: total the good and bad save levels separately, calculate them, add the result, but no). Then there were caster levels, they just stayed separate, and even if they hadn't there was no 1/2 BAB equivalent, so if you took a non-caster class, pht, nada. OTOH, feats and stat boosts were calculated from total character level and worked fine.
5e, perplexingly, looked at that, saw the problem, and fixed it for saving throws, attack bonus (proficiency in general) and caster level - then turned around broke it the same way for ASIs/Feats and Extra attack.
Why? Idontknow. He's on third...
It also, probably not coincidentally since MCing is optional, has classes that don't have anything like the 'building block'/modular 'elegant' design that the MCing system cries out for.
So. Yeah. My solution, so far, is to just not use MCing. I could see an extensive re-write of classes - or just tweaks of ASIs, Caster level, & Extra Attack - probably among other things, to improve on it...