This heads back into the problem of Schrondinger's combat and hitpoints
That is a problem with hit points, yes. They're a very vaguely-defined, abstract mechanic that models a mix of factors. So if you narrate
n hps to a character as a severe wound, you run up against a number of problems. The most immediate one is that the character is suffering no penalties for that severe wound. He runs just as fast, for just as long, hits just as hard, and so forth. The only effect is that he's
n closer to 0. Then there's how the 'wound' heals. If he has HD left, he can rest 1 hr, re-gain
n hps, and, if you're committed to the idea that those hps can only represent that wound, *poof* the severe wound is gone.
A similar issue happens as soon as you drop someone to 0. If you narrate it as a mortal wound, and then they up and roll a 20 on their next death save, you'll be back-peddling. Even if they stabilize and wake up 1d4 hours later, you might have an issue.
I have to explicitly avoid any description that involves obvious physical damage
That's one option. Another is to consider that there's more to hps than wounds being inflicted and healed. Take the 5e describing-damage sidebar, for instance. If a DM describes hp damage as inflicting physical wounds only at a certain threshold, then, clearly, there can be 'non physical hps.' Considering how many hps a D&D character can have relative to much larger creatures, it actually makes a lot of sense for /most/ of their hps, and the critical aspect of their hp maximum, to be based on those non-physical factors. What they are - endurance, morale, luck, divine favor, or whatever - doesn't matter, and could vary. So, in the above examples, instead of freaking about a severe wound being healed in an hour, you can assume that the HD rolled in that short rest just give the character back a bunch of non-phyhsical, say 'endurance'-based, hps. The wound is still there - it never gave a penalty anyway - but he's ready to dodge some more damage in the next fight.