Page 2 also says the following; "To hit the target, your result must be equal to or greater than the AC. If you hit, you deal damage with your attack, reducing your target’s hit points."
I believe it is the next line.
Funnily it doesn't define what a miss is. Presumably a miss is the opposite of a hit. So, "to miss a target, you result must be less than the AC. If you miss, you do not deal damage with your attack, and do not reduce the target's hit points."
Or in the Combat section (under basic attacks) p.19:
"If you hit, you roll damage, unless your attack specifies otherwise."
So, a miss is "If you miss, you do not roll damage, unless your attack specifies otherwise."
The rules say that "a hit deals damage". It does not follow from that that a miss doesn't deal damage. The most you could say is that it follows that a miss may not, or perhaps probably does not, deal damage. (On the "rolling" damage, I'll note that the ability under discussion does not involve rolling damage. It is fixed damage.)
In every circumstance a miss is a failure, not a half-failure.
That's not accurate. For example, many circumstances grant advantage, which is the functional equivalent of a reroll on a miss. The rogue at 20th level gets the "Ace in the Hole" ability, which allows turning a miss into a hit or treating a check or save as if a natural 20 had been rolled. And an evocation mage at 5th level gets the ability to do half damage even on a miss with (or an enemy's successful save against) a cantrip.
The greatweapon ability is, in my opinion, best understood as another form of dice manipulation ability along these lines.
I'm going to assume we are using YOUR version that the fighter tires the target out until they die. NOT Rodney's version that some part of the attack is so brutal it transitions through the armor of the target.
The ability as described by Rodney is one that is so brutal it cuts through the armor to do STR damage.
As I said upthread, these are just illustrative. They are not (and cannot) be definitive of every use of the ability.
Consider another fighter ability - Trip, on p 25 of the Classes document. If the superiority die equals or exceeds the target's STR mod, they are knocked prone. The giant ape in the bestiary has STR 22 (+6) and 95 hit points. Suppose a 1st level fighter hits the as-yet untouched ape for 1 hp of damage, rolls the superiority die and knocks the ape prone. How did s/he do this? 1 hp of damage to a 95 hp giant ape is barely a scratch - so how was the ape knocked down? Perhaps some fancy footwork by the fighter.
Consider the same ability used against a kobold. A kobold has 2 hp and STR 7 (-2), and I am pretty sure that most tables would narrate the same attack (1 hp damage, knocked prone) as some sort of winding blow that sends the kobold sprawling.
It is inherent in D&D's hit point mechanic that mechanically identical events get narrated differently, depending upon the fictional context and the external mechanical parameters.
why is it that a fighter is so unrelenting that he is able to tire a target out, even if that target is an immortal creature doing nothing more than dodging every attack, without also having that fighter also be tired out by wielding his greatsword.
Because the fighter is tougher than the immortal in question? Why is that a cleric's devotion brings divine intervention, but a fighter's devotion doesn't? Why is a back-alley rogue a better precision striker than a weapon-specialist fighter? Because we allocate different archetypical abilities to different character classes.
Now, outside of the ability saying that he is so unrelenting that he somehow deals the fighter's STR to the target every round they are in melee (or rather every round he makes an unsuccessful attack) what do you have to say that it SHOULD be this way?
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Outside of - this ability! - does anyone (non-magically, and yes excluding fate points if they would allow such a thing) have such a capability? Especially another fighter?
I've given examples above of comparable abilities that are part of D&D next.
As to why the game should include such abilities, isn't the answer that these are character types that players want to play? I don't know what other reason can be given for the design of character classes in a fantasy RPG.
(Why not just give a damage bonus? Because that (i) underminds bounded accuracy, and (ii) makes the fighter tread on the rogue's toes as the "spike damage" character class. Auto-damage, like advantage, conforms to bounded accuracy while still raising average damage per round.)
I am only barely aware of fate points (I'm including hero points, adventure points, or other variants). I've never used them as a player or allowed them as a DM. I know they can add bonuses or change outcomes, but I don't have any idea on what kinds of things they can change specifically.
Beyond that, as I understand it fate points are a rare and precious commodity. They don't replenish regularly and once spent they are drained.
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if they had fate points - a system I have never once used EVER - then yes I would let the fighter dictate something in such a way. I would do that because if I'm allowing fate points I'm allowing that kind of subsystem. It is not, however, a core mechanic of the game.
There are plenty of games that use fate points not as a rare & precious commodity, but as a core mechanical feature of the system. (Examples include Burning Wheel, Marvel Heroic RP, and as best I unerstand it Fate and Savage Worlds.)
Many people interpret D&D hit points as a form of fate point - in effect, the character pays fate points to "buy off" what would otherwise be the debilitating or fatal effects of injury (from weapon, fireball, falling a great height etc) - and on that reading D&D is also a game where fate points are not rare and precious commodities but core to the system.
I get it that you don't like the great weapon class feature. But it doesn't follow from your dislike that it is an inherently flawed mechanic. Having seen that sort of mechanic in play, I can testify that it does what it promises to do: it makes the character, in play, express the archetype of an unrelenting fighter who
will drive his/her enemies into the ground. If you don't want that sort of player-fiat in your game, that's of course your prerogative. But it is not unplayable.
the pixie is used to being small and buzzing around. He may have to buzz from place a little faster but presumably he is immortal and doesn't need rest or sleep (I believe those were requirements Celebrim put forth). Why SHOULDN'T the fighter be more tired?
Frankly, I am not interested in a game whose mechanics are designed around this sort of corner case.
Let's look at another monster - the Wraith. According to p 87 of the Bestiary the wraith is incorporeal, does not need to sleep, and has 78 hit points. A fighter attacks the wraith and hits it for 3 point of damage. The wraith now has 76 hit points left. What, in the fiction, has happened to the wraith? It has not been scratched - you can't scratch an incorporeal spirit! It has not in fact suffered any tissue damage at all - it has no tissue, living or otherwise. It has not become tired - it is an undead spirit that cannot tire and never sleeps.
Whatever exactly you think happens to that wraith - I personally think of it as pure meta-game, in that the player of the fighter has got closer to declaring, in fiction, that the fighter has defeated the wraith - I think the same narrative can be told about the pixie being worn down by auto-damage.
It is a fiat mechanic on the wrong side of the screen
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I assume you like fiat, I don't - it shouldn't be a necessary part of the core game.
It seems to me that this is the only issue. You don't like player fiat abilities.
Whereas I think they are inherent to D&D. There is no other way I can see of making sense of what happens when that wraith suffers 3 hp of damage and still has 75 hp left.