Since 5e is without built-in grittiness apart from PC death, it also does not make much sense to describe gritty healing.
HP simply is not flesh in RAW, you can roleplay it as being flesh but without any mechanical consequences.
Means, if you assume magical healing of HP aka not-flesh is painful, then, by this logic, taking HD within a short rest or healing total with a long rest should be especially painful.
HP is better described as stamina, including only minor bruises, maybe some hematoma and scratches or something like that, if you insist that dealing out 20 HP equals to a mutilation or your entrails hanging out, then you better check for infections also in the next step.
Did you know that the main reason for wearing armor was not to get hurt less during combat but rather to not get hurt at all? Even the most minor wound in medieval field conditions could mean a deadly infection, that is why.
HP mechanic reflects that very well imo.