I know, I know, the verisimilitude of hit points has been an ongoing argument for decades...
This is something I would like to try when I start up my next game: All hit point "damage" is simply small, accumulated weariness.
Damage is not broken arms, sliced open bowels, or arrows to the back. It's the blow to the shield that takes a little more out of you, it's the sore muscle from stepping out of the way, it's the stitch in your side. Nothing physical, nothing that could be permanent, even without magic, and that can easily be visualized to recover after a long rest.
When one is reduced to zero hit points and needs to make death saving throws... There, that's when there is actual damage from that last blow. That time the arrow hit you, the sword cut you open. That's what requires magical healing or multiple long rests to recover.
Has anyone experienced treating HP in this way?
No, I haven't.
It's a long ongoing debate that dates back to the 1970's. I personally believe that Gygax gave a fairly definitive treatment of the topic in the 1e AD&D DMG, but even there I have to concede that after appearing to answer the question he left the door open to other interpretations with a conflicting paragraph elsewhere.
There are several problems you are going to run into going with the "all hit points are not meat" definition. The most glaring of which is all the "on hit" triggers there are in D&D. For example, on hit you might suffer being poisoned. How did that happen if all hits involve no actual damage to the flesh? Likewise, on hit you might suffer energy drain or any number of other ill and potentially lethal effects. How does that work if you are just being fatigued?
When you say something like, "Damage is not broken arms, sliced open bowels, or arrows to the back.", you make it sound like there are only the two extremes here - "All damage is not meat" and "All damage is meat". I've never played D&D where the damage was broken arms, sliced open bowels or arrows in the back either. You've introduced a false dichotomy, and most interestingly, it's exactly these two extremes that Gygax definitively discarded in his treatment of what a hit point is. In Gygaxian D&D, hit points are some combination of meat and not meat, and the usual practice is that all hits (on PC's at least) do some combination of a wounds and depletion of whatever the not physical portion of the hit points as. The only real question is whether those things are strictly proportional. That is to say, the only real argument in my opinion is over whether a wound always has some physical component however minor, or whether the definition of "hit" allows for hits that are at least on some occasions entirely without corresponding wounds inflicted.
When on the other hand you write something like, "When one is reduced to zero hit points and needs to make death saving throws... There, that's when there is actual damage from that last blow. That time the arrow hit you, the sword cut you open.", that's per Gygax again entirely the way the game has always been intended to play. Even among the groups I've played with where every hit had some physical component, the grievous and potentially mortal wounds only occurred when you were reduced to less than zero hit points. Before that, you might have been nicked, cut, bruised, scraped, contused, pierced, strained and so forth, but none of those minor injuries were of the life threating sort. A particularly powerful hit might 'connect' harder than others and be described as a bad wound - an arrow sticks in your thigh for example - but it wouldn't be described as connecting with anything vital unless it dropped you below zero hit points.
Anyway, you wouldn't be running a game in a way that it hasn't been run before, but I don't think it is the way hit points have traditionally been described.