How it works in D&D, RAW, today is that a poisonous snake who bites you pierces you and poisons you when it does so. Which is why when you summon one and tell it to attack a skeleton, it's less effective than hitting it with your hammer - the skeleton isn't affected by poison and takes more damage when you bludgeon it.
If a snake bites you, or you hit a skeleton with a hammer, something related to "physical durability" happens. In the narrative, it's not just a near-miss or a drop in morale or a dodge that fatigued you. The skeleton isn't taking more damage because your warhammer makes it unluckier than your spear.
It ain't pure physical durability, but it involves physical durability, no "table rule" required. The blow also reduces the mental durability (it didn't like being hit), will to live (maybe it disrupted some necrotic bindings) and luck (the skeleton took a blow on the ribs instead of on its spine) of the skeleton, sure. But physical durability is a part of that damage, too.
If anyone accuses my game of magical elf make-believe of being "realistic," I want them found and silenced, hahaha.
For me, your last sentence makes an important point. The model is incomplete (well, technically all
models are incomplete) and contains contradictions. On the other hand, play is a symbolic and analogic activity: for many purposes it has to pass-off-as rather than be realistic.
Poison attacks are an example of on-hit effects. Here's the poisoning on-hit effect from Giant Spider:
the target must make a DC 11 Constitution saving throw, taking 9 (2d8) poison damage on a failed save, or half as much damage on a successful one. If the poison damage reduces the target to 0 hit points, the target is stable but poisoned for 1 hour, even after regaining hit points, and is paralyzed while poisoned in this way.
A poisoned creature has disadvantage on attack rolls and ability checks. A paralyzed creature has additional problems. Strangely, the poison spray
spell does not bestow the poisoned condition. As an aside, I like the use of poisoned and charmed in 5e to chain to other effects.
My general view is that at all times when playing RPG we're dealing with impressions of a non-existent situation. Each person at the table will have different impressions. Hopefully only slightly different, but probably - were it interrogated - containing both substantial differences and outright contradictions. We continually revise our picture in the direction of our interests and to match that of others. The impression - sometimes called shared imagined space - always, under every RPG system - contains a myriad of omissions and contradictions that we constructively fail to notice so long as they're not important to our play. On-hit effects are one of the known contradictions with D&D hit points. That's not new to 5e.
In this light, what counts as gritty realism or slow natural healing? It's not especially realistic. The healing isn't natural. In my home campaigns I have recently been sticking with an additional tweak - characters gain a maximum of six hit dice. One die per level up to level 6. Hit points aren't gained past that (not from Con, not from Tough). I've added a wounded condition, that like poisoned and charmed will chain to other effects (e.g. healing word
can't benefit wounded creatures.)
My concern isn't that the game feel more or less grittily realistic, it's that fights are palpably risky. I want players to not be too keen to get into one, and to know when they've been in one. If you've ever played the Bushido
RPG, you might have a sense of what this can feel like. I'm also aiming for something you summarise quite well:
That's part of why I made the point that the "grittier" rules are more about campaign pacing for me. I often like a slower, seasonal pace to my grand storylines, where weeks can pass in convalescence and downtime. Not always, but often enough that I'll use rules to help get that vibe.
The DMG and XGE provide a range of examples of what might be done in downtime, and I'm also interested in far more social interaction. Bearing a few wounds is unlikely to hinder a character making their way through social circles. Those wounds might even form a good conversation starter. It's all about purposes. Going back to the OP, "grittier" perhaps isn't a well-defined purpose. It helps to have in mind what grittier means. It might not mean realistic. So far as GR and SNH go, it readily leads to something more aligned with a more leisurely campaign arc, with more going on in downtime from adventuring.