The problem isn't "WAR is too good". It's more the Tank Jobs' leveling are all over the place.
WAR gets Raw Intuition in the 50s (56), after which point it has a lot of control over its own health pool. Even before that, its -3 (26), Thrill of Battle (30), and just after Raw Intuition, Equilibrum (58) help it out a lot.
DRK gets the same heal at 26 WAR does, but doesn't get a lot of its heavy tools until later. In the case of Oblation in particular, much later (82).
GNB is overall more middle of the road, and Aurora isn't extremely fast acting and comes at 45, but it also has the -2 of its combo giving it a heal and shield from level 4, which is the earliest healing access any tank has. GNB's mitigation kit is also well balanced, with its personal mitigation not having limits like DRK's does (magic only).
PLD has the strongest mitigation kit, but despite it's motief as holy knight, and contrary to popular belief, is actually the tank that gets its sustain LATEST. PLD's rotation DOES NOT HEAL until level 84 when they get Divine Magic Mastery II. Until then, PLD has no healing in its rotation. None. No HoT, not heal, no barrier outside of Knight's Benediction from Holy Shelltron which...they got 2 levels earlier at 82. Yes, it gets Clemency, but not until level 58 (2 levels before PotD cap), and that requires them to stop their GCD rotation for a damage loss as they GCD heal themselves with it. It's also not AT ALL sustainable until level 64 when they get Divine Magic Mastery (1) that halves the MP cost of all their spells. At that point they can cast 5 Clemencies in a row and refill MP with Riot Blade (1-2 over and over), but before 64, that's 4,000 MP for a single Clemency, which isn't sustainable in any way.
But at 90, the tanks are mostly balanced, even in dungeons. WAR's easy to heal, sure. But so is PLD and GNB. DRK is less easy to heal, but not HARD to heal (oh no! the WHM actually has to cast CURE TWO instead of unstopping Holy/Glarespam?! The HORROR...!), and is still easy to heal if it IS using its mitigation suite correctly. WAR is braindead, but that's what it is/does. PLD and GNB don't have the problems DRK does at 90 dungeons, so clearly it's a DRK problem, not a WAR problem. And the problem is that DRK has more mitigation but doesn't fill its own health pool as much. But how many DRKs are willing to trade TBN or Oblation for Equilibrium? Probably not many, since the place those tools are most useful are Savage and Ultimate raids, which is where WAR is balanced evenly with the other tanks. The damage gap isn't huge (and as others have said, DRK was the uncontested leader from 6.0-6.4 itself due to how well its kit works with downtime and phase transitions).
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Ultimately, all tanks can deal with dungeons fine. Any that can't are buffed, but everyone can. "But my healer has to actually press a heal button!" is a feature, not a bug. If anything, that should be more common, not less, but 4 man/casual dungeon content "doesn't matter", so it is what it is.
For leveling, the issue is, as I say above, when people are getting abilities, not some horrible imbalance at the level cap. But this is the way most MMOs work these days. They only really care about balancing level cap content. As long as things work "well enough" (and they do) while leveling, it's a back burner issue.
I think we'll sadly never go back to those days.
Because the playerbase behaved badly at the time. In the pursuit of the easiest path to rewards, people benched classes and insisted various Jobs were "bad" because they needed to work with others. That's why stuff like Esunaing Berserk went away, and why all the "piercing/slashing/blunt/etc damage up" debuffs went away.
The payerbase just can't be responsible with those things and complains about them endlessly, so the devs don't want the headache of trying to keep them in when min-maxers will complain about them.