Maybe the length of a "long rest" can depend on whether the damage is superficial or deep?
Most of the damage is intangible: loss of energy, alertness, and luck. Only if reaching zero hit points can the damage become deadly.
The character is fresh at full hit points. Down until half hit points, there is some contact, but the hit point loss comes from fatigue, getting sloppy, etcetera. At bloodied, at half, the damage causes lasting bruises, cuts, bloodied nose, and so on. Bandages become helpful. But still the damage is cosmetic. One can look at a fight sport match to see what this looks like.
These superficial bruises etcetera clear up in, say, 2d8 days. This cosmetic damage has no bearing on hit points, but it is fun to remind the player, the character still has a black eye from when getting bloodied several days ago.
But at zero hit points, ones luck runs out. The damage becomes deadly. This is the sword thru the gut, and so on. (The attacker can choose avoid lethal damage, but the target is still unconscious and completely vulnerable at zero hit points.) If the damage is deadly, and the character survives the death saves, this deep damage can take weeks (or even months) to heal.
Typical injury − broken bone, bad burn, deep cut − 3 to 9 weeks to heal.
So, when does real damage happen? It can happen at zero hit points. It normally takes weeks to heal real damage.
Consider the following rates of healing. It might be too complex for gaming purposes, but it gives a sense of what injuries might look like.
Damage Categories
• maximum hit points until half: negligible injury, mainly fatigue, heal normally
• half hit points until zero: minor injury, bloodied, heal normally, cosmetic marks persist 2d8 days
• zero hit points, fail 0 death saves: minor injury, downed, heal normally, cosmetic marks persist 2d8 days
• zero hit points fail 1 death save: moderate or serious, 2d6 weeks to heal
• zero hit points fail 2 death saves: severe or critical, 3d8 months to heal
• zero hit points fail 3 death saves: untreatable, permanent loss of either life or limb (DMs discretion based on injury)
The game doesnt have to get too complex about damage categories. But switching from an 8-hour sleep for full hit points, to something that takes weeks, can make sense when an injury is significant.