Involved TTRPG Combat
I just wanted to give my own two cents on "involved" combat encounters in D&D and similar TTRPGs.
Generally give your players the expectations that combats are something to be solved rather than beaten. Communicate that with them clearly in order to create more fulfilling combat that isn't hitting things until no enemies are still alive.
Give combat encounters a goal. I'll list a few examples later, but generally you should think of what an encounter is meant to achieve for the players. Sometimes it's fine if it's just "an obstacle meant to be overcome", but this shouldn't be the only goal.
Consider tipping points, when it becomes clear that victory is certain for the player characters, but the combat could still continue. Create off-ramps for such encounters, where you can have a satisfying narrative way of ending it earlier. Fleeing minions after the leader has been killed is a classic.
Consider the environment. Cover, different types of elevation, and dynamic elements (vehicles, rotating gears, a rushing stream, etc) will make an encounter more engaging. This is more challenging to run theater-of-the-mind than with minitiatures/VTTs, but still doable.
Use different types of enemies in the same encounter. A simple way is to use both ranged and melee attackers, allowing your players with ranged options to pick off the enemies further away, while the melee-based heroes have their own enemies to deal with.
Have enemies with multiple phases. In 5e, enemies with the Mythic trait (mostly found in Mythic Odysseys of Theros and Fizban's Treasury of Dragons) are one example for this, as they gain more abilities the more damage they take. Alternatively consider creating monsters that consist of multiple stat blocks, where defeating certain parts of it changes what it can do and what your players can do to it. E.g. turn Mazaal from The Legend of Zelda - The Minish Cap, into a D&D monster.
I'm not good at making monster stat blocks, but imagine a construct with two hands (each with a distinct stat block) and a head that shoots beams from its eyes. The head has a Tethered trait that makes it invulnerable while the hands (that can grapple your heroes!) are still active. Once the hands are defeated the head starts jumping around like crazy, knocking over the heroes with shockwaves as it hits the ground. That sounds like a very engaging combat encounter to me!
Now when it comes to combat goals, here are a few suggestions:
Ensuring the safety of an escorted NPC or a valuable item that the party is transporting to a different loccation.
Keeping an enemy in a magic circle that allows it to be killed permanently, since it would otherwise just regenerate or be outright invulnerable.
Waiting for reinforcements as hordes of enemies attack the party, placing a timer on the encounter.
Preventing an enemy from reaching a specific spot on the map, such as an altar or a mechanism.
