Not wading through the rhetoric; just responding to the OP's request.
Why do I find damage on a miss believable?
First of all, Mike Mearls has clearly stated what hit points represent: "Hit points represent an element of physical wear that involves a combination of fatigue and physical injury. As you take more damage, you have more evident wounds"
So, the question can be restated thus:
Why do I find a combination of fatigue and physical injury on a miss believable?
The reason is because I have personally experienced it. I have fought at martial arts competitions locally and nationally, and absolutely it makes complete sense to me that avoiding an attack (and thus causing it to miss) will increase my fatigue.
In fact it is also true that I have suffered physical injury on a miss. I assume that if an attack is blocked, then it counts as a miss, and it is absolutely true that I have seen physical injury -- bruising, torn muscles, that sort of thing. I think I once got a broken rib when a blocked kick smashed by fist into my chest. I know I entered the fight without a broken rib, and only dropped a point, so odds are I suffered the break on a "missed attack"
Sadly, I have even been seen people defeated by misses -- one guy so exhausted and bruised up he dropped to the floor and was unable to continue.
If you have the view that a miss is something that never causes damage, then by definition this mechanic is not believable, but enough other stuff in the game causes damage on a miss that you cannot make that general statement for D&D. It's pretty clear that, speaking very generally, damage can occur on something classified a miss.
D&D is a mostly binary system. Criticals and Fumbles blur the line a little, but most modern games (and quite a few old D&D spells) have a more smoothed approach where you can fail, but have a little effect. That is all this mechanic is -- a failure to do a significant hit, but it still wears out the opposition a bit. Given that I have seen that happen a lot in real life, it definitely seems realistic to me.
If you want something unrealistic, a much easier target is the fact that a guy with a dagger doing d4+0 cannot possibly kill someone with 30 hit points no matter how good an attack he does and no matter how bad a defense the other guys does. Given that HUGE level of unrealism in combat, damage on a miss seems a pretty minor thing to worry about.