Wanna disclose that the only ttrpg I've played in depth is dnd 5e, so other systems might offer interesting answers.
TLDR I want to make combat more interesting as it progresses, not less
So combat in dnd is what should be the coolest and most entertaining part of a story, but is often the slowest part of a session. Most of combat is spent waiting for your turn. When it is your turn, sometimes, you'll swing twice with your sword, miss twice, and that's it, that's all you can do. Even when you hit, the consequences are often just an invisible number going down. Not very interesting, there's next to no input from the player, and this is mostly just the dice deciding everything. No room for roleplay or storytelling here.
So how, from a game master's perspective, can we make combat more interesting? A straightforward solution is to just have a bit of story content in each turn. Describe a fighter missing their attacks as "you are locked in combat with a warrior, who narrowly blocks your blows. The sound of steel on steel rings through the battlefield." Doing this for every turn is exhausting for the DM, where they have to try to give a flavourful description for everything, but every now and then can give more vivid images to your players.
Dialogue is another way to insert storytelling into combat. I've seen no DMs ever enforce the "6 seconds of dialogue per turn" rule in dnd, because it just sucks. Have the antagonist exchange barbs or shout their ideology at the players. Have them discuss their past with the player. Describe it as them shouting at each other over the wind, or the sound of war around them. Again, this can't be inserted into every turn, or it too will become monotomous.
So what about mechanical ways to elevate a battle? Legendary actions in dnd serve this purpose, to allow an NPC to perform actions when it isn't their turn. This helps to alleviate the action economy problem and makes the NPC seem a lot more active and dangerous. If a boss can attack when you don't expect, it makes the mechanics fade back into the background a little bit as your players realise how powerful this character is.
I think debuffs are the most frustrating thing to happen to players. Being able to do less without any long-term change to your characters is just annoying. Getting disadvantage on an attack means your character is less impactful in a session. As a player, this sucks. Imo, debuffs should be avoided unless they either apply to everyone fighting, including your enemy, or they advance a character's story. My DM actually achieved both of these scenarios. They designed a combat encounter where difficult terrain was cast by an opposing spellcaster, and their fighter and ranger could navigate difficult terrain easily. This made us realise that the enemy had planned their attack to our specific environment (forest) and that they were particulary dangerous in this specific location - but if we meet them again under different circmustances, they will lose their advantage. In another encounter, one player, who's character has been lacking control of their own life, was suffering massive debuffs from a character who was trying to control their mind. They had to make saving throws every turn, but the stakes were a lot higher than just missing the next attack - losing a saving throw could permanently change their character.
Debuffs are hard to pull off, but buffs aren't. Imo buffs are the easiest way to escalate a fight - have your NPC become stronger and more dangerous as a fight goes on, rather than them losing resources like health and spell slots. Have your NPC become stronger after losing a certain amount of health, or even have a second and third phase with different attacks and new descriptions - this makes a boss fight feel much more tense. You could also give your player a weapon that becomes stronger after landing more hits or something, or an accessory that halves their hp and gain advantage on every attack. This makes them feel like they're becoming cooler and more powerful as the fight goes on, too.
Are there other systems that better escalate combat? I find that combat in dnd becomes more predictable the longer it goes on due to the system of health and spell slot attrition. Characters in a fight only lose resources, but don't become stronger at all. A lot of power fantasies have fights become bigger and more bombastic as they go on, because that's fucking cool, but that doesn't happen by default in dnd unless you try to make it happen. So do other players or game masters, or anyone with experience in other systems, have anything to weigh in on? My ideas are just ideas and I haven't actually tested these, so I would love to hear from others.
this ended up being a lot more text than i intended.
5e probably has the worst combat of a game that isn't a meme. Movement basically doesn't happen and the only real option is some form of attack, and it's still slow as hell.
From what you're asking, I want to suggest PF2e, which does actually have interesting combat and monster abilities that can change the flow of battle, but I think what you actually want is a more story telling style system like one of the Powered by the Apocalypse or Forged in the Dark systems. PF2e combats certainly can get more tense as the fight goes on, but isn't specifically geared towards that kind of escalation, whereas story telling systems naturally lend themselves to rapid, flavourful, escalating combat.
FWIW, I think D&D combat just kinda sucks, which is a problem because it's obviously the biggest presence in TTRPGs. I'd say it's because it's both mechanically simple without much interesting decision making happening (especially so for martials) and takes way too long. It would require so much work to fix these problems that you might as well just play a different system. If you want a simple escalation mechanic, 13th Age (which is inspired by D&D4e) has the "Escalation Die", a d6 that increases by 1 at the end of every round. The number on the face of the die is added to all attack rolls made by all combatants.
For larger changes, imo there are generally two ways you can go, and these depend largely on the group and GM:
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Make combat more interesting by adding mechanical depth. Get extra crunchy with it, to the point that the pseudo-boardgame is actually fun as a turn-based game. Instead of "I attack the orc twice", which is simple and boring, make the thought process into "okay so that orc has less armor on the hands so maybe I could try to go for the wrist to make him drop his shield. But should I feint first because he's using a tower shield that will make it easier to block? Or maybe just ignore that and forgo my defenses to make two swings at the neck and hope one goes through?". Games that go this way are things like GURPS (running tactical combat with extra options from Martial Arts enabled), or to get really cronchy, The Riddle of Steel (or its successor Blade of the Iron Throne).
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Make combat more interesting by largely eliminating mechanics and focusing on narrative stakes. The characters meet the BBEG, they both talk back and forth for a while, and then when words are over and it's time for a fight, you resolve it with as few rolls as possible, perhaps even a single roll. I have less experience with this side of gameplay tbh, but as I understand it games like FATE or most PBTA things tend to move more in this direction.
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The best combat systems I play aren't combat systems. They're scene resolution systems.
Games like Hubris Games' Story Engine have you do RP, make declarations of how you're planning on doing things in an adjudicated scene (battle or otherwise), sometimes spawn cut scenes to establish aid for the main scene, etc. and then roll the dice once to determine who wins/loses and how well/badly they win/lose. Then you RP through the consequences and proceed. Story Engine was, to my knowledge, the first published game (as in not just articles in A&E at the time) that completely dropped the wargames side of RPGs in favour of story-focused game play. It wasn't perfect, but it sure was an eye opener.
Games like Theatrix ditch wargaming roots as well, along with dice or other randomizers, in favour of what amounts to an economy of Plot Points (rather similar in usage to a rudimentary version of what FATE does now with Fate Points). It is the flow and ebb of Plot Points that resolves scenes (combat or otherwise) and guides the story being told forward. Again, it is flawed, but flawed in interesting and important ways that you can learn from. (It is painfully obvious to me that the designers of FATE studied and learned from Theatrix.)
Games like Spark use a kind of hybrid mix with an added cycle. Scenes progress by players making Bold Statements that are treated as having happened unless challenged by another take. Once challenged, a system of gathering points vaguely similar to Story Engine's approach, combined with the Plot Points/Fate Points approach of Theatrix/FATE resolves which take prevails and the scene continues from there until certain exit conditions are met. Again it doesn't matter what kind of scene it is: combat or something else.
I'll try to add to the conversation with 5e rather than suggest another game like everyone else (although I will say I've gotten into 4e recently and it's surprisingly fun with very tactical, engaging combats lol). I agree with a lot of what you said about sprinkling in descriptions and dialogue throughout the combat. I also think better designed enemies can help a lot. I highly recommend the Flee Mortals book by MCDM. I used it a lot in 5e campaigns now. More interesting enemies can make comabt more interesting.
I also think more interesting environments can also help. Put in lots of cover, spots of difficult terrain, maybe a trap, things like that.
Uninteresting combats should be fast, and one thing that helps that out is low health, high damage enemies, which for me means every now and then I lower enemy health if they already got a few good hits in or an enemy blew a spell slot to take care of them, since the point of this combat is to make players lose resources, not really for the story. This is less of a problem for low level combat, but more for higher level combat when health totals increase. Also, for these don't be afraid to make them some run to warn other allies, or run in fear if they get low health (unless they're mindless undead or constructs), or some run when their boss dies, things like that.
On the other hand, interesting or story important combats I'm okay with being long as long as they're also compelling and fun to play. After all, D&D has a lot of fun combat options in classes and players want to use them. That's where I recommend things like villain actions in that Flee Mortals book or the other stuff I recommend above.
Oh and the last thing I did is I am very liberal with consumable magic items or temporary buffs as awards for quests and stuff. I like the latter because players aren't encouraged to hoard it and it in fact encourages pushing ahead more to take advantage of the buff while they have it so the opposite mentality of hoarding, but the former has the nice advantage of being an additional option for a player to use in combat, which can make it more interesting, especially for those without spells. But to prevent hoarding, you have to make combat harder and have the kinds of players that look at everything on their sheet, as well as give enough rewards they know they'll get more in the future.
Play something other than D&D. It's the bottom of the barrel, impossible to get worse without picking something like FATAL.
I have way more of a martial arts background than a ttrpg background, and one thing I've always wanted to try is being specific about incoming blows and rewarding quick decision. For example, the DM says "The raider's sword is arcing downwards at your head" and if the player says something within a second or two --"I block","I parry"-- them they get a bonus to that action. I feel like it could be a fun way to inject a little quickness into turn-based combat, and encourage other players to have their next move ready.
Generally this comes under the whole "you don't have to be strong for your character to use a sword" issue, but if you actually want to parry attacks you want a PF1e swashbuckler.
