I'm snipping out a few bit - trying to do so where I agree or it's a supporting point I'm already addressing to keep the size reasonable. If you feel I took out something more please call me on it.
Again, I can see systems where this might work - Amber Diceless is one I've already mentioned. If you want to run a combat in a system where the combat outcome revolves around narrative currency, and the participants work together to narrate the combat, then a 'three act' rule for a combat might be fine - opening, rising action, climax. Two skilled players could probably produce something like the duel from "Princess Bride" under these conditions.
I fully agree, these would be the best types of systems to run something like this in, and in a more traditional game it might feel tacked on or imposed. Because it is. But that doesn't mean it can't open up new mechanical avenues of support just like Inspiration tried to in 5e. Let me address that after your next part.
But fundamentally, you've "reframed" this by attacking my movie metaphor and other details, yet I don't see you really threatening my central point - which is that a combat in an RPG continues to be exciting so long as the fiction continues to meaningfully evolve.
I feel that I have addressed it, but I left it in the movie metaphor and didn't really explore what it means in a RPG setting. It is a point orthogonal to your very valid point you make about the encounter evolving.
Often we talk about alternate win conditions for battle besides attrition. Save the commoners who are going to get sacrificed. Stop the ritual. Deal with the stirges while trying to escape the cave-in. Stop the necromancer from escaping while his undead minions try to slow us down.
Commonly these are seen as "better" or at least more interesting encounters than just attrition where each side is attempting to kill each other, and the side standing at the end is the winner. (Who can then heal up for the next encounter.)
Now imagine that every non-climatic encounter was like this. But with win (or partial win) conditions either way depending on the tide of the encounter. Even standard attrition type battles can be reframed like this - instead of having to kill each and every bandit, as long as you've won two or three of the rounds the others will surrender and you can take them back to justice. But if you've lost the majority of rounds then perhaps their reinforcements arrive and you are captured.
Again, this isn't saying what you are talking about where an encounter with bandits is going badly and then the PCs rally and turn it around isn't a fun game. It definitely is. And that encounter is not static. And the Vampire suggestion, where if everyone is having a good time to extend more rounds, would allow that to happen. But if there's no dramatic turn around and there's an obvious trend on what's happening, then let it end in a dramatic way.
Heck, having reinforcements show and capture the PCs is just a start of RP, escape attempts, bribing or converting guards, etc. A grind of a battle the PCs are losing can end up in a TPK.
Marvel Heroic Roleplay codified this, where the DM can spend 2d12 from the Doom Pool to end a scene in the villian's favor.
But that doesn't mean it needs to end. If the encounter is still tense and the PCs have a shot (if losing) or something to lose (if winning), then continue as long as the players are having fun. Vampire's option does not seem about forcing a stop, from that description it's about putting in an escape valve for if it is turning into a grind.
Again, I am agree with everything you say about "tense, exciting, fun to imagine, producing dramatic moments, or shining moments of awesome" encounters and keeping them going.
If you have an encounter that is going to go 3 rounds, and then already be a grind, then there is definitely something wrong with your encounter design that arbitrarily ending the battle won't fix.
At 10-15 minutes for all players and the DM to take their actions, at this point we've invested 30 to 45 minutes in this scene after three rounds. Most other scenes would be long over with that much table time.
The need that combat takes more wall time than anything else,
and should be allowed to do so is a legacy holdover. If it's exciting and the players really want to continue do so, but otherwise move onto the next.
As a bit of my personal soapbox, D&D and like games make everyone good at combat while allowing the game to function with only 1 or 1-2 competent PCs in other pilalrs (how many trackers do you need? how many lockpickers do you need, etc.) is an artifact of how long combat takes, and it's a self fulfilling prophesy with combat-focused character creation/advancement mechanics and long combats feeding each other. Another game that doesn't want that same dynamic can easily look at making combat scenes the same length as other scenes.
Perhaps if you wanted a plot twist, escape, or other bit of storytelling, you should have designed possibilities like that into the encounter in the first place, rather than going, "Gee... three rounds have gone by... this is a drag. Perhaps I ought to invent on the fly some away the fiction has meaningfully evolved."
But in reality, HOW OFTEN do we hear, see and experience DMs running an attrition combat? It's by far the default. Making something else the default
up front (so it's not in the slightest the "Gee" line you put up there, but something planned from the beginning) helps lead to more interesting combats from the get-go.
Fights get grindy when you have uninteresting terrain, uninteresting goals, and uninteresting foes that do the same basic thing round after round, and winning is a matter of simply eroding hit points without making any real choices.
That is one time fights get grindy. They also get boring when they are just about being the last person standing 90% of the time, then you heal up and do it again with little real meaning outside the combat, no matter how interesting it is. They also get annoying when they take up 2 hours of your 3 hour play session for a non-climatic battle and nothing really gets accomplished except you're one room closer to the end. Things drag on when sure it's an interesting fight but you want to get to the oracle so you can get the answers to who killed your father THIS SESSION. It's a drag when combat ends out the session so the DM can't lay down a satisfying ending nor a cliffhanger. It's a falsehood to say that only poorly designed combats can drag on too long.
This is saying "let's intentionally force the DM to design more interesting stakes for combats, and one that has an escape valve if it isn't staying interesting enough after some time has passed".