I'm in favor, in principle, of introducing a generic martial mechanic that can be gained and utilized in vastly different ways, while maintaining a common "interface" to a wide variety of different effects.
The key thing is to make sure each class speaks to this interface in a mechanically distinct and flavorful way. Clearly there is plenty of middle ground on the issue, because we don't usually grouse about everyone using a d20 but there was plenty of grousing about, e.g. AEDU (and I mention it without judgment) for being too homogenous. The rest of the thread has covered the arguments pretty well, if not to the point of consensus. The common spell list with different ways of learning, regaining, and casting spells is, I think, a decent example of a middle ground I find acceptable.
For example, maybe the main fighter schtick is getting the expertise dice renewed automatically each round. That would make the fighter an incredibly reliable source for the things he does, whatever they are. What if a rogue gained ED (they should be more careful

) based on tactical consideration, and possibly varying with schemes? This would let us get away from "every rogue sneak attacks" to something more flexible. Here are some rough ideas (not intended to all work at once)
- Gain the dice if you have advantage against at least one enemy at the start of your turn. This makes "sneak attack" quite a bit more dynamic, especially because the party and the enemy both have a say in whether it happens.
- Gain the dice at the start of your turn if you are threatened by only one enemy. (Duelist, essentially.)
- Gain the dice if you were not hit since your last turn ended. (Fits various highly defensive, mobile, and/or stealthy conceptions)
A warlord (as a separate class, or just as a different mechanism for fighters) might be based around granting ED to other characters, which could work quite smoothly if they are a generic resource. For example, it might be a much smoother way to grant allies damage or protection instead of a bajillion interrupts, because every character can at the very least use them to increase damage or reduce damage they take. They might refresh their own supply based on more strategic considerations, and maybe even starting combat with a larger-than-normal pool, to represent their tactical genius early in the fight.
- Refresh when an enemy falls.
- Refresh a random amount at the start of each turn by making some sort of "commander check".
- Refresh when an ally gets a critical hit (more of a "rah! rah! go team!" warlord)
A barbarian could use them to power a rage mechanic that is a lot more fluid than n/day or n/encounter, relying on what actually happens in combat. Unlike other martial combatants, a barbarian might accumulate these dice over multiple turns to unleash truly monstrous effects, or choose to spend them as they come in if the situation calls for it. They could refresh based on what might actually make a barbarian angry
- Gain dice when hit (revenge)
- Gain dice when allies are hit (revenge for friends)
- Gain dice when the barbarian misses (frustration)
- Gain dice when anyone crits or is killed (bloodlust)
This could work well with multiclassing. For example, a fighter5/barbarian5 might have 2 ED from fighter levels, maybe just 1 from barbarian (but with an accumulation mechanic) and mix and match the possible uses from either class as desired. Heck, it could lead to the simplest version of Tenser's Transformation ever seen: Gain ED as a fighter of your wizard level.
In short, we have barely started to cover what is possible with the basic idea of ED, and there is plenty of room to explore outside the "refresh n dice every round" idea demonstrated in the playtest fighter.