Request for concrete features
If you think I’m wrong, then I need the specific support or control features you believe materially change that outcome so I can model them. Broad categories like “support” or “utility” aren’t actionable, I need the actual features.
That is one path that can be explored to make the analysis feel less like white room theory crafting and more like a real game. Another path is to clarify what the actual scenario is. Currently, it feels like the scenario is both competitors are slugging into a punching bag for 6 seconds and some scientist is measuring how far back the punching bag is swinging.
Actual scenarios we could consider are:
- The Barb 6 and the Fighter 3 / Bard 3 fight one another. As I alluded to in an earlier post, in that scenario, the True Strike's radiant damage pierces the Rage DR, which cuts the Barb's eHP in half, and if the Bard also casts Bladeward, then their eHP goes way above the Barb's. I think there's a strong argument here that in a duel scenario, the multi-classed pretty boy beats the single-classed brute. Is it the most relevant scenario? No. But it's an interesting one. Certainly, if in-game, while role-playing the Barb said "we all know you are abysmal in combat and only good for playing your flute!" and the Bard threw down the gauntlet and then proceeded to beat the Barb fair and square in a duel, and then proceed to heal the Barb after making them unconscious, the Barb would probably shut his mouth and respect what the Bard brings to the table.
- There is a swarm of rats. Each has 1 HP and 10 AC. A character with more attacks wins over one with fewer attacks, no matter how strong they are, and one with an AoE (such as Thunderclap) wins even more. Again, advantage to the Bard if they have that cantrip.
- There is a mob of weak monsters like goblins. The dynamic is similar to the rats, but minimum damage counts for more. The Thunderclap's 2d6 (with 6th character level scaling) has more likelihood to leave a goblin alive. The Barb probably has an edge here but it's slight, and depends on how surrounded the characters are. If the Bard is surrounded by 8 goblins, then two rounds of Thunderclap on those 8 has a decent shot of killing them all, which would then get us a rate of 4 dead goblins per round, which the Barb will have a hard time competing with at "just" 2 attacks per round (who's not scaling now?). Maybe the Barb can compete via TWF or Cleave, but either way, it's still pretty close.
- A boss fight. One strong monster. Now things like damage over time and Vex weapons start mattering, whereas they did not really up until now. This is the scenario where the Barb's 2 attacks and higher DPR really shines (though Heat Metal as @Maxperson said could be significant, assuming the boss carries some metallic gear).
- A boss with minions. This is a mix of scenarios 3 and 4 at the same time, but with the extra twist that if you charm the boss, he could walk away / let the heroes walk through, and the minions would just follow the lead of their boss. Persuasion and Charm Person are not going to work in every scenario, since it depends on starting attitude, skill checks or saves, and other factors, but I think it's fairly clear that the Bard has an edge here... Of course, you could also say that when the boss is killed, the minions have some likelihood of having a morale failure and rout, in which case the Barb is the best since they will kill the boss the quickest. But charming the boss costs one slot and no HP, while killing the boss costs one Rage and probably quite a few HP, so there is also a degree to how much you can win an encounter.
In the above 5, I'm giving the win to the Fighter/Bard on the 1st and 2nd, and the win to the Bard on the 4th (i.e., I'm assuming the boss carries no metallic gear). The 3rd one is IMO close to a tie but maybe there is a slight edge for the Barb assuming the characters are probably not
totally surrounded by 8 enemies each. The 5th one is the hardest to adjudicate, if the DM allows persuasion or charms to work, then the Bard has the most efficient way to win the encounter, whereas if it comes to blows, the Barb can slay the leader and rout the minions sooner. You can craft any number of other scenarios. But the point is, there are a variety of them.
Ultimately, it's clear that the party with both heroes is way ahead of a party with two Barbs or another with two Bards. So in that sense,
the strongest build is the one the party doesn't have yet.
You say "straitjacket," I say "strong thematic cohesion."
I get why (some) people enjoy 3.X-era-style talent trees, but neither system is superior in an objective sense. It's all just taste.
I am like a cat chasing a laser pointer with the above tangential discussion, but I really should be focusing on this which is much more on-topic

...
Yes, it's ultimately a matter of taste.
However, I don't see why feats would have lesser cohesion than the class/subclass system. I believe your point is that you pick one feat from some theme or path, and then pick another feat from another theme, and that in your view lacks cohesion. Sure. But assuming your table plays with the optional multi-classing rule, which most do, then you are always free to not keep going in a class and start leveling another one. If you do so, then how is that anymore cohesive than picking a martial feat at 4th level and a magic feat at 8th level? It's not... either system allow for branching out into another direction.
The reason I say subclasses are like straitjackets is that with feats, I can pick Elemental Adept (fire), then for my next feat I can pick Elemental Adept (acid). Whereas if at Wizard level 3 I pick Illusionist, then at level 6 I must pick the level 6 Illusionist feature. But why am I railroaded into that subclass, why not allow rebooting into another subclass?
It would be nice if, for example, at Wizard 6 I could pick the Diviner level 3 feature and thereby start another specialization. And then at level 10, I could pick yet another level 3 Wizard subclass feature (and thereby dabble in a 3rd school specialization) OR the level 6 subclass feature of either the Illusionist or the Diviner. I don't see why it would be thematically inappropriate for a Wizard to go to medium depth into both Illusion and Divination (which are really two sides of the same coin... make naughty word up, and see through said naughty word). A Wizard that splits their specialization in two is still much more thematically coherent than a Wizard who multi-classes into a Barbarian (ok, granted, this is the worst possible combination

but it's just to make a point... more plausible combinations like Fighter/Wizard would still be a bigger thematic departure).
There could be balance reasons why a Wizard 6 with the 3rd level features of two subclasses and no 6th level subclass feature might be too strong. So I'm not necessarily saying that this subclass reboot system I just described is actually desirable. 5e was not designed for that so it carries certain risks. But with feats it is much cleaner to support these options of either going deeper into a very narrow path, branching out slightly into a different but still closely-related path, or going farther afield into a completely different path. Whereas with the class system (in the absence of the above hypothetical subclass reboot mechanic) you have only the 1st or 3rd option, not the 2nd one.