The fact that the TV shows do not ask you to sit through stretches of recover is irrelevant. If a character on a TV show was shot out in the woods and curled up to sleep under a tree at the end of one episode and woke up the next morning, at the start of the next episode, fully recovered, that would be stupid. The fact that it was "off-screen" has nothing whatsoever to do with the problem.
This is all a degree of selection. The game is "mute" about narrative in all editions. It has always been up to the DM and the players to make the "narrative" their own, if they so choose.
I'll give a 3.x example to illustrate:
You can have a DM that tells you, "Roguey McRouge that hit just knocked you on your ass. The gash left on your side is bleeding (take 10 points of damage and you're prone)." Or you could have the one that said, "Hit, take 10 points of damage and you're prone." Game effects are the same whether I describe them "narratively" or not.
After the fight ends, the rogue character has no healing potions, and he doesn't have a heal stick. So he decides to make camp. He curls up under a tree to rest. The next morning he wakes up and has full hit points. How is that? He had a bleeding gash and hit points damage, did his "wounds" all of a sudden close up? No, he's a 10th level character and after a rest he regains his level in hit points. Whether that "wound" was described as "physical" or not is irrelevant, after a nights rest he regains his level in hit points. No magic, no narrative connection to how his "wounds" are not there anymore.
According to your definition that would be stupid.
According to my definition it makes perfect sense and can be easily supported by the "narrative". "Off screen" the rogue binds his "wounds", stretches his muscles, and takes some food before curling up under the tree, the next morning he is still sore, but his "wounds" won't affect his main purpose, "Finding the Eye of Rah."
The reason is that the "wounds", a term the DM is providing not that the mechanics are forcing on us, are part of a "pool" of resources called Hit Points. Hit Points represent all kinds of things that let you keep on fighting (physical and "metaphysical"). This has ALWAYS BEEN the definition of HP. In mechanical terms HP has never had much to do with "health" in the classical sense.
The disconnect I'm seeing when people complain about this issue is what I'll call selective reasoning. I can find a "reason" for magic to heal "wounds". I can find a "reason" for my level to heal "wounds". But it is a bridge too far finding a "reason" for SPENT healing surges to heal "wounds".
In 4e, if a character is "wounded" in battle, DM terms, and he spends his second wind, his "wounds" don't all of a sudden disappear. Just like the wounds of the 3.x rogue didn't miraculously disappear the next morning. That character is "narratively" digging deep and using reserves of "guts" to keep fighting. Mechanically he simply uses his second wind, SPENDS a healing surge, and keeps fighting. That healing surge is SPENT until an extended rest. His "reserves" are depleted for the rest of the day.
To take it one step further, "narratively" you have even more options in combat. When the cleric uses Healing Word you can describe it as actually closing wounds because it is "magic". When the warlord screams at you with Inspiring Word and tells you to, "stop sniveling, and get back in the fight", you can describe it as the shame of not measuring up. You dig into your reserves and keep fighting. At the end of the fight you take a short rest to bandage wounds, stretch your muscles, and catch you breath. You SPEND healing surges to do all those things. Then, you are ready for the next challenge. You've used up "reserves" during that short rest. Resources that are no longer available for the remainder of the day. You want to have "wounds", that is what SPENT healing surges can "narratively" represent. It is up to you, just like the "wound" was "narratively" up to you.
At the end of the day you settle to make camp, you treat your "wounds", clean out your bandages, take some food, get a well deserved rest. In the morning you are still sore, but it won't impede your main purpose, "Finding the Eye of Rah."
After the adventure and as part of the campaign, the DM can "narratively" describe the long recuperation period if he so chooses. The game rules do not impede that, the same way that they do not impede the DM and players coming up with their own narrative for the "wounds" in the first place.
That is why I don't have a problem with the narrative at all.