Among the points I'm looking for in an RPG damage system are:
1. Avoiding death spirals.
2. Avoiding requiring a player to be "the healer".
Death spirals, seen for example in many Shadowrun editions, is where it gets harder and harder to avoid death such that characters can go from hurt to dead much quicker than they go from unwounded to hurt.
The first levels of exhaustion make it hard to recover from a string of bad luck and could qualify, but I'm more interested in exhaustion level 3. Given that there are other ways to gain exhaustion, including failing the Mortal Wound recovery save during a long rest which changes the long rest from reducing it by one to increasing by one (a 2 level of exhaustion difference), this is a real possibility. And at level 3 you now have disadvantage on those death saves, making it that much more likely to die.
As for requiring a healer, I've played D&D for a long time and remember the resigned "ok, I'll play the cleric" of early editions. One of the improvements made in the game back in 4e edition was that you didn't need a healer. One was fantastic, but it was possible to adventure without one just like it was possible to adventure without any other specific niche.
The restriction of HD recovery to 1-2 per long rest removes this evolution. Yes, it allows a longer term attrition battle against characters, but the amount of self-healing only vs. a single healer now available over let's say a week of active adventuring is staggeringly different. This is an active step back, removing an evolution in gaming systems.
I see what you are going for, and I'm can support a table with lasting consequences and long-term attrition. But the these two points are of real concern to me.
The first could be addressed by allowing exhaustion to be recovered on a long rest regardless. If you're wounded and rest, you have rested.
The second I'm not so sure what to do. Magical healing makes the HD recovery reduction minimal, but makes playing without magical healing a much bigger penalty then playing without some other character role. Personally, since the affect is not a big deal at all when you have a healer, I'd just drop that and allow the normal half-HD recover per long rest.