To make that perfectly accurate, I think this: Any class with medium armor proficiency can wear medium armor and be a suitably effective adventurer - given the right ability scores, medium armor can even be the best choice, even if the character is proficient with heavy armor.
This is true.
Problem is, the "right" ability scores are trending towards scores that aren't best in class for other purposes, so few players (with an eye to efficient charbuilds) will consider using medium as more than a transitory state.
The complaint is that medium armor isn't a minmaxable end state in itself. For it to be such a state, there needs to be a game statistic, let's call that Grapefruit, that you can minmax and then use both for defensive and offensive.
Such game statistics are offered for light armor (Dexterity) and heavy armor (Strength). The first one you can use for finesse and range weapons, the second for regular weapons. There does not exist a Grapefruit statistic which, if optimized, leads to best-in-class AC and another significant game usage (such as offensive).
For instance, let's consider the situation if Wizards were given medium armor and medium armor could be used with Intelligence. Then you have a situation which would satisfy the OPs complaint. You have a competitive build (the wizard) which can put everything into Intelligence and use it both for AC and for something else (not weapon damage in this case but spell DCs are certainly a competitive option).
The reason I say that the barbarian class is built around not wearing armor at all, and medium armor proficiency is the "peripheral" trait, is this: the unarmored defense feature and d12 hit die are parts of the character at level 1, and the starting equipment choices in the class description do not include armor of any kind, meaning for many groups that it is owning medium armor which might not happen until some higher level.
I understand this, but I would say barbarian unarmored defense is kind of an exception.
Yes, most level one abilities are part of the core identity of characters based on that class. But there are exceptions, and I would like to suggest this is one such exception.
I would even go so far as to say most Barbarians run around in medium armor (again assuming players with an eye on efficiency; I am well aware you can ignore a point or two of AC to run around unarmored), especially considering how most groups skew towards lower levels.
Plus, assuming you are using the starting gold rolls at the beginning of the equipment chapter instead of the class description +background equipment options, you have to spend 50 of your potential 80 gp on armor to match or best what having a 16 constitution (not uncommon for barbarians, according to my experience) does for the character.
The cost argument does have a point, but I would reserve that point for fighter plate mail. 50 gp is in comparison nothing. Even a single adventure should give you enough money to buy whatever non plate armor you like.
If you can have a 16 Con that's great; but I certainly hope that's not your highest score, because while having a good Con is excellent for every character (and I certainly recommend no character to stick to a mere 10 in Con), your best score should go to Strength, so you can dish out some hurt.
I won't go as far as saying you're wrong, but AC 16 does require Dex 16 if you have Con 16. And still AC 16 is too low to get a passing grade in the long run (at the first few levels you obviously have no choice).
And that's simply a non-minmaxed state. Generally you'd be better off either leaving Dex at 14 (so you won't "waste" any points while wearing medium armor), or you should go all the way up to 20 (striving for unarmored AC 20).
But that's the general case, which pretty much assumes dumping Strength. Since barbarians can't (and indeed, shouldn't - believe me, I've tried) do that, they're much better off shooting for Strength 20, sticking with Dex 14, and gunning for the best Constitution they can get while still not delaying getting to Strength 20 (feat analysis not included).
And so we've come full circle: if you can get Con 16 under those circumstances, great, but if you roll badly at chargen (or don't roll at all) I don't see that happening in general.
So while I am (again) not saying your case can't happen, I still expect most Barbarians (again, run by efficency-minded players) to use medium armor for the bulk of their career. It's simply the cheapest way of fixing your defense so you can focus on your offense.
And, to get back to the original point, that this makes the Unarmored Defense feature of Barbarians somewhat unlike, say, Rage, or - say - Action Surge, or even Monk Unarmored Defense, in that it's significantly less core (again, from a minmax, or even efficiency-leaning, perspective).
Barbarian Unarmored Defense is best viewed as a luxury ability; one which is great for roleplaying and style purposes, but isn't fundamentally
necessary - you can stick to medium armor for your entire career, since after all, medium armor can eventually get you to AC 20 (+3 half plate), which I think should be considered a bare minimum AC at high level. Sure Bracers of Defense gets you AC 22, of course, but if you have Strength 20, Con 20,
and Dex 20, I can only congratulate you (I can't expect you).
Best regards,
