I don't see how it's stretching the fiction to say that a PC recovers because of an inspiring dream, or seeing something inspiring through half-closed eyes.
For me, it's because it's the martial power source. The martial power source should be martial. In a sense, non-magical.
The inspiring dream should be psionic or arcane or even divine.
If the "seeing something inspiring through half-closed eyes" is allowed in the game, then this non-magical inspiration should occur regardless of who above you is trying to inspire you (similar to Intimidate, anyone should be able to do it).
If Charisma can be used to fight good with a melee weapon, then it should be able to be used to encourage an ally to no longer be unconscious through "half closed eyes" by anyone.
In other words, if the narrative description is that the unconscious PC is being inspired, then any PC should be able to inspire that PC because it is not a magical or psionic dream.
Bottom line, the mechanics drive the narrative instead of the other way around. (Non-magical) healing by shouting does not make sense, so the narrative has to be changed so that it isn't actually healing so that the narrative can fit the mechanics. Otherwise, it doesn't.
And let me turn it around - how can a game like Runequest or even 3E replicate either of the scenes I mentioned - Aragorn recovering with a dream of Arwen, or a fallen fighter recovering from a wound/swoon as s/he first sees, and then hears, her leader calling upon her to get back up to her feet.
The first scene can be handled via magic easier than through non-magical descriptions, even in earlier versions of the game.
The second scene doesn't matter. It's not important. The only reason we are even talking about it is because of the illogic of non-magical healing in the game resulting in a plethora of narrative changes to what healing and hit points means.
I have never once played with a PC Warlord in the group where the narrative described "seeing through half closed eyes" was used. Instead, it was always hand-waved away. "You're healed".
I've also had a player (and separately, a DM) say "Warlord healing is magical" as part of the PC's narrative and background description, just because non-magical healing from across the room is so jarring to some people.