I think the multiclass character gets the proficiency because it's a specific domain feature rather than a base class feature.
Correct. From PHB 164, "Proficiencies" heading (emphasis added): "When you gain a level in a class other than your first, you gain only some of that class's
starting proficiencies, as shown in the Multiclassing Proficiencies table." "Starting" implies the inherent and universal features of the class, not specific special things only acquired through subclasses. Otherwise, you'd get horrifically wonky things, like (say) a Sorc 1/Bard 3 being unable to get Martial Weapons and Medium Armor from Valor Bard.
And yes this is one (minor) failure of the otherwise best ever multiclassing rules.
[citation needed]
More seriously, "best" multiclassing rules is even more subjective than most rules discussions. I don't actually have a preference myself--I like aspects of both 4e and 5e class-mixing, but neither one is even close to "best." I'm a huge anti-fan of
à la carte multiclassing just in principle because of the system mastery it invites both "positively" (careful selection can be broken-OP) and negatively (often "punishes" choosing purely for flavor*)--same for 4e's hybrid rules, albeit to a lesser extent purely because they came relatively late rather than being the core MC method. I'm also not a big fan of the "MC feats are INSANELY GOOD, but don't bother investing any further" thing from 4e--particularly since Paragon MCing is, itself, another "trap" most of the time.
*This is why I value really rigorously balanced math in any game I play--be it an RTS, a TTRPG, a 4X, an FPS, whatever. When the options are sufficiently balanced such that the mechanical advantage for some choices instead of other choices is too small to make a practical difference, people are free to choose whatever they like, confident that the game will
support their choices. It ceases to be a competition between flavor and power, because power doesn't change enough to be worth bothering about, and thus flavor becomes the determining factor.
I think they were too conservative with these exceptions, it would have been simpler to just "get the best weapon/armor prof from all classes", and it certainly would not have been overpowered.
Well, part of the reason for it is to avert one-level dipping. Cleric has the unfortunate interaction effect because their solution to "some clerics wear heavy armor and others don't" was to make it a subclass feature. No solution to the proficiencies question can capture all of "simple, different,
and zero interaction effects." The method they went with is simple, and lets different subclasses have different proficiencies, but fails to avert interaction effects. Your method is essentially the same, it just chooses a different interaction effect. For example, going with your method means making Fighter a highly, highly attractive one- or two-level dip for any character that wants to mix it up in melee. My Bard, for instance, could've skipped Valor Bard entirely and gone with Lore Bard instead, and come out barely the lesser for it. (Of course, part of it is simply that the Bard capstone ability is
crap so getting Bard 19, or heck even Bard 18, is nearly as good as the full class, so a one- or two-level dip worth the equivalent of three+ feats is an insanely good deal.) With the method they chose, you now have to decide whether low-level Cleric benefits + Heavy armor is better than low-level Fighter benefits + medium armor. Even with all my complaints about the Fighter class, I don't know that we can say that the Cleric is automatically the winner.