I can play a relentless fighter without having any such ability available. It's not like an fighter who rolls below the target's AC is "relenting". There's no connection between having a relentless attitude towards life and being able to deal a small amount of damage when you roll a 1 on your attack.
Actually, there is. That's the whole point of the mechanic. You are relentless, therefore, even when you roll a miss, you still deal damage. It's not like it's a really complicated narrative here. And, note, the "towards life" thing is something you're adding, not what's being talked about here. The warrior is a relentless combatant - constantly pushing forward in combat.
Which is what this mechanic is modeling.
Are there other ways we could do this? Sure, no problem. Extra attacks, bonuses to hit or damage, rider effects on a hit, that sort of thing. Each has its own pros and cons. This is one way of doing it.
But, let's not get too carried about building this mountain shall we? We're talking about one fighter type. Not all characters. So, right off the bat, this may or may not be seen in any given campaign. And, in all likelihood, it's one option out of several that can be taken, not an automatic.
And, given that you, Ahnehnois, make such a big point about owning the game as a DM, I would think that removing a pretty minor rule like this, which has virtually no knock on effects, would be a fairly simple choice. Additionally, trying to marginalize it by proclaiming that it's only for a small subset of playstyles doesn't reflect the fact that the poll is pretty much split down the middle. Should we ignore half the players so that the game favors what you want? Isn't that what 4e did, and you complain rather bitterly about - ignoring the player base?
This is a pretty simple rule to adjust to taste. Take ownership of your game and adjust it if you think it's a problem.