Obviously if this ability is in play then damage can be dealt without hitting on an attack. At the mechanical level, the player has a "fiat" ability - s/he can simply declare an attack and thereby bring it about that, in the fiction, the target is somewhat worn down. In the fiction, the fighter in question is so unrelenting that every 6 seconds of combat with this guy guarantees that the target will be on a trajectory towards death.
I understand that. What you fail to understand is that this is backwards. You are starting at the mechanic and proceeding to derive the explanation. You aren't really overly concerned with whether the explanation is consistent, whether the ability is appropriate for every instance of the character's positioning within the fiction, or whether the mechanic is justified by a need to model something within the fiction that was so valuable to the fiction and such a well known trope within the genera that its exclusion constituted a failure of the model. It's enough that the mechanic is interesting and has been declared by fiat to be a part of the fiction, and you'll rearrange the fiction as necessary to incorporate it.
But presumably you are familiar with the idea of games with fate points, which players can spend in order to fiat outcomes rather than rely on the dice? In that case, it might help you to think of this ability as building into the fighter a fate point resource for the player - every time the fighter attacks, the player has a little fate point s/he can play which says "No matter what the dice say, you take at least 3 (or 5, or whatever) damage from this attack".
But what is the fate point modeling? In this case, sometimes it is modeling hitting the foe extra hard, and sometimes it is modeling being exceptionally accurate. The first might have something to do with 'great weapons', but the second doesn't. And don't fate points normally model luck, or the universe rearranging itself in your favor because 'it likes you' or because you have control over it? You aren't really concerned with why the fiction requires fate points at all, only that they have some mechanical effect. If this is a fate point, shouldn't the mechanic be tied to the charisma or luck or wisdom of the fighter, and two handed weapons should have nothing to do with it? Really, this has nothing to do with fate points at all. The only relationship to fate points is exactly where I'd expect you to be focused - on their mechanical resolution within the game. You aren't interested in their similarities within the fiction, because the fiction is quite clearly not where your priorities are at.
The fighter does have a capability to deal extra damage - namely, this ability! And why should the fighter tire him/herself out quicker than the pixie. The point of this ability is to model an unrelenting fighter. S/he strikes so swiftly and viciously that the pixie gets worn down, and loses the energy to dodge, more quickly than against an ordinary foe.
But if that is what it models - losing the ability to dodge and tiring an opponent out - why does the model rather abstractly and obtusely model this as hit point loss, changing the entire history of damage and model of hit point loss for the sake of this one mechanic! Why not model this as applying a penalty to the pixie's dexterity bonus, or adding a 'fatigued' condition to the pixie? Why is this model tied not to 'unrelenting' (a claim I find spurious), which implies a Constitution or will based power, but instead tied exclusively in the model to 'overwhelming', large weapons (that are tiring to wield), and high strength? This is incoherent, as the explanation you are offering for the fiat judgment doesn't even match the fiction of the mechanic or the rules of the mechanic! And again, it is a model of hit points that no edition of D&D has ever fully embraced, and certainly none before 4e even toyed with.
I already answered this - every time you roll a damage die you have to vary your description based on both the base hit points and the current hit points of the target. The need to vary your description of what happens when you deal damage on a miss based on the circumstances of the target, and the ingame context, is no different.
It is different. This time, you have no physical injury involved at all! This is outside the realm of the usual variation. I can't think of a time I've ever narrated to the character the opponent failed to make contact with you at all, but yes, you just lost 38 hit points (or even 1!). This new possibility creates all sorts of incoherencies. Suppose for example the target isn't slain, but instead loses 55 of 60 hit points entirely to 'hit on a miss', never having been made contact with. If the target was merely fatigued, why does it require so long for the target to regain full fitness? Ahh, but in 4e we totally remodeled healing so that it made some sense to say that hit points where just fatigue... even though no prior edition had said that and each modeled fatigue in some other way than hit point loss. But, if it is fatigue, why don't 30 seconds of vigorous aerobic exercise - running up stairs or dancing or sprinting - also do 55 points of hit point damage to you?
No one ever got killed by a scratch, or even a cumulation of scratches.
For certain definitions of scratch, sure they have. An accumulation of small wounds might result in sufficient loss of blood to lead to unconsciousness and from thence death. Collapsing from blood loss and to be bleeding out happens all the time in the fiction of my game. Presumably, the colossal equivalent of 'mere scratches' leading to blood loss and shock is the main way characters kill enormous monsters. For a colossal monster, all the monsters hit points might be meat!
The rule makes perfect sense. It is a fiat mechanic...
It's a fiat mechanic in a different way as well. It works, because you say it works. The mechanic says something happens, so it happens and the fiction alters itself to conform. The mechanic isn't conformed by the fiction, and you feel no need to make it conform. Again, it's really simple to make this rule conform to the fiction. Instead of saying 'damage on a miss', you say 'damage in cases of making contact with the target that would otherwise not normally do damage'. Note that unlike 'damage on a miss', this later mechanic is pretty exactly what you've most often described the mechanic as modeling. Damage on a miss doesn't in fact model what you've largely narrated in your attempt to rationalize it to the fiction. Now damage is consistent, it's tied logically to great strength and physical force, and it's clear to see what is happening. Most accurately, this involves figuring out which misses made contact, which involves breaking the normally abstract AC into its less abstract components, but if this is too much of a chore then you could claim 'damage on a miss that doesn't miss by much' is a reasonable approximation. It wouldn't be perfectly congruent with the fiction, but I could see it as a compromise between ease of play and simulation.
Of course those who like process simulation mechanics don't like it, but that tells us nothing about its coherence as a mechanic.
Sure it does. People who like more process and more simulation don't like it precisely because it is incoherent. What do you think their disagreement is over? I don't necessarily mind that you are willing to accept high degrees of incoherence in your model for the sake of say tactical interest or simplicity of resolution, but let's not pretend that you aren't prioritizing whatever you are prioritizing over coherence.
The real puzzle for me, as always, is why people who like process simulation mechanics would play D&D rather than (say) Runequest or HARP.
Ok, the real puzzle to me is why people who don't like process simulation mechanics insist on hijacking my game of D&D and trying to turn it in to something else. Why was it necessary to kill my preferred system? Couldn't you just get your own? If you don't like alignment, hit points, levels, Vancian magic, clerics, etc. etc. and you were never really happy with D&D before 4e, why do you think I should be happy with your butchering something I was happy with?