Game time is only meaningless if time passage in the campaign is meaningless. There are concerns involving supplies and the activity of monsters in the area which have meaning.
OK, I'll put it another way: game time is only as meaningful as the players (including the GM) make it - and that is all. Moreover, it is only relevant in the ways that the players want it to be relevant. If that way doesn't specifically feed into the strategic importance of resources and healing then linking healing and resource replenishment to game time won't help add strategic elements.
If the environment is not reactive then yes, passage of game time is pointless.
It doesn't have to be just reactive - it has to be specifically set up to demand action constantly as a tension to the time taken to heal. If the tension isn't constant, the game time is, for our purposes, meaningless. To get that tension needs some sort of invented reason for hurry.
Pre-4e non-physical damage is usually a status effect. Stat damage/drain, stunned, fatigued, shaken. These are not healed with cure spells.
Pre-4E?? Pre-3E there were no such things as stat damage or statuses except 'petrified'. For most of D&D's history, damage, be it physical or non-physical, does = hit points.
I, personally, don't have a problem with the possibility of morale based healing as long as it's not as powerful or prevalent as magic or skill based healing.
Sigh - back to the old saw of "magic should be more powerful than not-magic" again...
A feat to 'man up' and do an adrenaline based self heal in combat for a fighter would be fine. Although if it's adrenaline aren't temporary hp a better model?
Aren't all hit points temporary? I'm not looking for a scientific break-down of what the healing does - just an instinctive, non-rational understanding of what it represents.
But frankly it makes more sense for that sort of thing to fix status effects (like fatigued/shaken, not so much petrified) than to fix gaping wounds.
"Gaping wounds" are things that only dead adventurers and NPCs have in D&D.
In "real world" terms it doesn't "make sense" that faith healing can close gaping wounds, but in D&D we aren't in the "real world". Who knows what magic is inherent in the body of a Hero?