But this is the bit that I don't understand. In this case, why can't the hp recovery be recovery of the capacity to go on despite the physical injury?
The biggest issue is that it makes the physical trauma disappear from the model, such that the model no longer reflects the reality.
A good way to visualize the old model that I (and those like me) used during 2E and 3E is to think of two separate tracks. The first track is your maximum HP, which represents your personal capacity to withstand physical trauma - how much of a beating you
can take before you fall unconscious. The second track is the damage you
have taken, which is how beaten up you
currently are. Unconsciousness happens when the second track extends beyond the first track. (You could also think of HP as a bucket, and damage as water which can fill that bucket, with unconsciousness happening if the water overflows the bucket.)
The value of your maximum HP represents a combination of different factors that aren't really worth tracking individually. These are your skill, your actual physical toughness, etc. This number goes up when you gain a level, or when your Con score changes, or you are under the effects of certain spells.
The amount of damage you have taken is only that one thing, though. It goes up when you suffer physical trauma. It goes away when that damage is healed, either naturally over time or instantly through magic.
Inspirational healing (including things like Second Wind) attempts to change the first thing (your capacity to withstand trauma) by instead changing the second thing (the amount of trauma you are currently suffering), under the logic that the only thing we
really care about is how those two values relate to each other. And that logic just doesn't hold, because some people actually
do care about whether someone is hurt-and-fighting-through-it or not-hurt-at-all. Those are two vastly different narrative states, and a good model
should be able to distinguish between the two.
Aside from that conceptual issue, the most obvious gameplay problem occurs when you look at the characters deciding who to heal. With our traditional model, they can look at someone who is beaten up and immediately determine that they need healing. With inspirational healing, they can look at someone who is beaten up and
not know if they need healing, or if healing would even
do anything. An injured character who has been inspired up to full is incapable of benefiting from healing magic, because their wounds are strictly cosmetic at this point.
For RP purposes, you might still want to heal those wounds anyway, but we don't have any information about how bad those wounds
look, since the rules of the game - our language for understanding how the world works - all agree that the character looks just fine. It also creates a conflict between the
character who wants to help their friend, and the
player who wants to conserve resources in order to increase the chance of survival, and it's never a good idea to exacerbate the gap between player and character.