Using your fists is not unique amongst fighters at all especially not in League.
I went ahead and looked through the Champions who use their fists in fighting, including those who use a weapon besides. I probably missed one or two, but my count comes up to around 15. 15 Champions out of nearly 150~ now.
- Alistar (no weapon)
- Blitzcrank (no weapon)
- Braum (door)
- Galio (no weapon)
- Gnar (boomerang)
- Lee Sin (no weapon)
- Maokai (no weapon)
- Rakan (no weapon, can't count a mirror)
- Shyvana - (claw-gauntlets)
- Sylas (chains)
- Udyr (no weapons, can't count primal magics)
- VI (gauntlets)
- Volibear (no weapon)
- Warwick (no weapon)
- ZAC (no weapon)
But what makes Sett unique is that he's a Juggernaut whose kit is dedicated to his bare-knuckle brawling. He's not a crowd-controlling minotaur, not a magically-powered golem, not a folk hero, nor a giant, magic-absorbing statue...Etc.
He's a guy who fights with his fists. No real magic to him.
Pit fighter who is also a momma's boy may be unique, but that's because it is a stretch and seems forced.
What? Okay, so what about other Champions, then?
Kayn being able to resist - and even potentially overpower a millenia-old ascended god-warrior through sheer audacity is a stretch, and it seems forced.
VI having a change of heart mid-heist and joining the Piltover police because of newfound morals is a stretch, and it seems forced.
Viktor entering a depressive state at how human beings favor emotion over efficiency and shun moving towards mechanically-assisted perfection is a stretch, and it seems forced.
You could say literally all of League is a stretch and it seems forced.
So if that's going to be your catch-all counterargument to something genuinely unique, then frankly, there ain't much left to discuss here.
Kayn and Zed are Ionian as well and inherently crave violence, being master assassins and a part of an order of killers.
Zed don't crave violence, though. Zed despised the Kinkou's inaction in regards to Noxus invading and ruining Ionia, and consequently set out to fight the invaders his way, since the passive nature of the Kinkou would only lead to Ionia's destruction.
Kayn has his own motives - personal perfection and superceding his master. And more than that, he's not some sloppy butcher, he's an assassin, and a ruthlessly efficient and calculating one, as shown by his color story.
And besides, Zed and Kayn don't paint a picture of all of Ionia - they paint a picture for themselves. Just because one small group wants to take the fight to Noxus doesn't mean all of Ionia condones the shift from peace to war - though through Sett's bio, it's clear that there's a latent hunger for violence and bloodshed across the land.
Assassin != craves violence. Zed doesn't crave violence, but he knows that it's a necessity to drive back Noxus. Kayn similarly understands the necessity, and uses death (and the fear of it) as a weapon. Both of them use violence as a way to protect and empower Ionia.