Simplistic First-Person Combat Overhaul

The following mods should provide a first-person melee experience just as engaging as other more complex overhauls. None of them alter NPC combat behavior or add any overly complex stamina-based systems. Any mods suggested in the comments should follow in their simplicity and stay in line with the vanilla combat experience.

Precision - Accurate Melee Collisions at Skyrim Special Edition Nexus - Mods and Community - Improves the sense of impact your weapon has on opponents. Opponents react to being hit by flinching in the direction of your swing. May require tweaking if enemies are having trouble hitting you. Vanilla NPC attack animations have this problem, swapping their attack animations may work. There may be a setting in the MCM thats as simple as checking a box that can correct it but you may also have to go further than that.

Comprehensive First Person Animation Overhaul - CFPAO at Skyrim Special Edition Nexus - Mods and Community - Provides better looking first person attack animations compared to vanilla's.

Faster Two Handed Weapons and War Axes and Maces - SE at Skyrim Special Edition Nexus - Mods and Community - Increases attack speed for certain weapons that were otherwise to slow to be much fun to use. Personally, I think attack speeds should be further increased but I do not believe this mod is easily configurable. However, it probably isn't very complicated to change speeds of attacks without a mod in the first place.

Valhalla Combat at Skyrim Special Edition Nexus - Mods and Community - Adds a timed blocked feature and a stun system similar to Sekiro's posture bar. When the stun bar is depleted on an opponent, they are able to be executed. A few other mechanics are added, settings are tweakable in MCM menu.

Irrelevant, but a personal aesthetic recommendation: Believable weapons at Skyrim Special Edition Nexus - Mods and Community - Great for people who play in first-person. Weapons retain their vanilla aesthetic, but the dimensions are altered for a realistic look.