%HP damage is not an unhealthy mechanic
There's a lot of complaints about %HP damage, so I want to explain why it's important and necessary for the game. There is a list of every spell that does %HP damage at the bottom (reason why the post is long). I'll only list damage scaling, and no healing. This post is about why it's okay to have damage scale from enemy HP, especially max HP. For the same reason, I won't list any damage that's amplified based on how much HP they have, if it doesn't scale off the target's HP. Examples would be Morgana's W (Tormented Soil) and Veigar's R (Primordial Burst) which does up to double damage if the enemy is low HP, but scales based on their own AP, not the enemy's missing HP.
Now, for the argument. #Point 1: %HP damage, generally, means the damage output is relevant throughout the game, and the champion is effective at fighting tanky opponents. There are obvious exceptions, such as Zed's passive. He's not very effective at fighting the frontline. But look at champions like Amumu, Dr. Mundo, Mini Gnar, Gragas, Jarvan IV, Kled, Kog'Maw, Malzahar, Nasus, Poppy, etc. They can do significant amount of damage to enemy tanks or fighters.
#Point 2: %HP damage encourages building tanky over damage, and targeting tanky opponents over squishy opponents. %HP damage simply doesn't scale as well as flat damage by buying damage stats (such as AD and AP). This encourages champions with %HP damage to build tanky, so that they can do more damage by using the %HP damage more, instead of oneshotting squishies. They also do more damage to frontline than backline, and encourage them to fight frontline instead of always attacking the backline.
#Point 3: HP stacking needs a way to be punished. When facing mixed damage, without %HP damage, stacking HP is an easy way to gain effective HP. This needs a way to be countered, like armor stacking or resist stacking.
In short, %HP damage makes a champion build tanky but still do relevant damage to enemy frontline, all without having a ridiculously high flat base damages. %HP damage means these champions usually can't oneshot squishies, and must fight the frontline. This isn't always true, and for newer releases, the exceptions have other reasons. Older releases also generally have a kit that doesn't really go well together, so not all %HP damage is going to fit my descriptions. But %HP damage in general is a healthy mechanic that should be kept, and even be introduced to older kits.
The list
W (Despair)
P (Blaze: initial DoT and explosion)
W (Tactical Sweep: outer cone only), R (Hextech Ultimatum: %current HP on-hit)
Q (Infected Cleaver: %current HP)
W (Parallel Convergence: passive has %missing HP on-hit)
Q1 (Neurotoxin: %current HP), Q2 (Venomous Bite: %missing HP)
R (Agony's Embrace: %current HP)
P (Duelist's Dance: %max HP true damage), R (Grand Challenge: same as passive)
R (Demacian Justice: %max HP true damage to villains)
W (Hyper)
W (Drunken Rage)
W (Harsh Lesson: not the tentacles, just her AA enhancer)
P (Martial Cadence: %current HP)
E (Thundering Blow)
P (Whisper: %missing HP on 4th shot)
R (Super Mega Death Rocket!: %missing HP)
W (Sentinel: the passive soul-mark proc)
P (Mark of the Kindred: stacking %current HP)
W (Violent Tendencies), R (Chaaaaaaaarge!!!)
W (Bio-Arcane Barrage)
Q2 (Resonating Strike: %missing HP), R (Dragon's Rage: %bonus HP of the kicked target)
R (Nether Grasp)
W (Twisted Advance)
R (Children of the Grave)
R (Fury of the Sands: damage aura)
Q (Hammer Shock)
W (Flail of the Northern Winds)
E (Two-Shiv Poison: %missing HP)
Q (Twilight Assault), E (Shadow Dash: %bonus HP)
E (Fling)
P (Glory In Death: on-hit damage after death), W (Soul Furnace)
W2 (Regurgitate)
R (Subjugate)
W (Blighted Quiver: when proc'd)
W (Silver Bolts: %max HP true damage)
W (Denting Blows)
(live) Q (Hungering Strike)
(rework) Q (Jaws of the Beast)
R (Crescent Sweep: %current HP)
R (Maiden of the Mist: %max HP when attacking the same target)
W (Unstable Matter)
P (Contempt for the Weak)
Honorable mention to Ziggs, who kinda has a %HP damage with the tower execute, I guess?
), and only 2 of those (Vayne and Vi) have %max HP damage.