Is it simply that the attack cannot miss? Is that the major stumbling block here? I think it is, but, I'm willing to be corrected.
No, it's not that simple.
For example, if you instead had an ability that said something like "every time you attack an opponent, any creature within that 5 ft square takes damage equal to your strength modifier in addition to any damage the weapon might deal", you wouldn't have the same questions of logic that DoaM raises. You probably wouldn't have a viable or balanced ability either, and some might argue that a condition or saving throw should be applied to this damage, but you could at least describe it as area damage within a 5 ft square. It might not meet everyone's scrutiny, but it would radically change the conversation and plausibility and internal logic probably wouldn't be the focal points.
As it is, regardless of how one conceptualizes damage on a miss, whether it involves physical contact or exhaustion or indirect wear and tear, the distinctive aspect that violates the d20 paradigm is not that it occurs on a miss, but that it somehow does not occur on a hit. The character has apparently acquired the ability to "relentlessly" deal damage only when his normal attack fails. Somehow he realizes that his attack is going to miss and his relentlessness kicks in. That makes no sense.
If you had an ability that said "instead of making his normal attack(s), the fighter selects three consecutive 5 ft squares; every creature within those squares takes damage equal to his weapon die plus his strength modifier, Dexterity save for half", it doesn't raise plausibility questions. It actually makes me think of the opening to Gladiator. Again, there's room to debate over balance; where this ability lies in the character's advancement scheme, how many squares, how much damage. But I think the basic approach holds alright and doesn't raise the same plausibility questions (though one might still debate that it takes a lot longer for an axe stroke to happen than an explosion).
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Similarly, an ability that granted a (presumably qualified or limited in some way) ability to take 10 on attacks does not raise the same plausibility concerns. Of course, such a character would still miss sometimes over the course of his career, but situations would arise where the character could not fail; only if the task was relatively easy, and the character was skilled enough to declare the attack trivial. This is perfectly fine, and even desirable, rather than rolling every attack in mopping up an easy battle. It's also consistent with other implementations of the take 10 mechanic and its use during stressful, high-leverage situations. I've implemented it as a high-level martial ability and no one raised an eyebrow. It's the sort of approach I'd like to see expanded as it provides a way to represent skill without further inflating spine numbers (which is supposedly a goal of 5e).
In this case, replacing the die roll with guaranteed mediocrity duplicates the DoaM function of automatically dispensing with weak opponents without worrying about a 5% chance of a fumble, while avoiding the DoaM corner cases where a DoaM character of overall modest skill is somehow able to damage extremely well-defended opponents. It's also worth noting that as currently constituted, 5e characters/monsters still has relatively low ACs relative to PC attack bonuses, and a lot of them would be hittable in this scenario. I find that players often enjoy narrating their high-level fighters rampaging through weak opponents without having to roll attacks; in my mind, it's no more implausible than a high level rogue using Skill Mastery to sneak by the guards every time. However, the rogue can't sneak by everyone so easily, and the fighter can't kill anything so easily.
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Which to my mind is why all of this "fighters can't have nice things" is BS. There are many nicer things than damage on a miss as it's being discussed here, which can not only duplicate relevant aspects of the DoaM functionality, but do a lot more as well.