Here's an example of why Yasuo, Zed, Yi, Ekko, LB are either banned or played in nearly each game.
It's not exactly tankiness as it is. There're tankier champs.
It's is not damage. There're Darius, Sion, Nasus, Renek and other tanky champs with lots of damage.
It is not even mobility: Rek'Sai is mobile and Aurelion Sol too...
It's damn rapid snowballing — that's what makes them monstrously unbalanced in SoloQ.
TL;DR: early kills let you dominate too strong.
I am attaching an example of a game played today. We lost 5 vs 4.
I should be ashamed of it, maybe, but I am not. Not much really.
After 4 or 5 good kills and assists Diana annihilates virtually anything in sight. Oneshot: BANG! You're dead, she's snowballing even further.
At this stage of battle only whole team together can stop her. If you have but one player being away — you just stay away and let her push, or you get killed.
What an ADC 4/0/1 can do? He deals a ton of damage, true. Only he needs usually some time in order to land all his shots.
A good fighter? Still vulnerable to crowd control and damage too.
Mage? It depends, but mostly you still need survive yet.
But when it comes to Diana, it's: Q, succesful hit, dash forward, oneshot! Next escape.
And the same is true for most of the assassins.
It is extremely hard to hit a dashing champ with a skill shot --- unlike ADC or mage.
He deals his damage so fast , you have almost no time to react.
He usually can escape.
At start this is counterweighted by damage sustained: Zed simply can't oneshot a healthy enemy.
But once an assassin takes enough feeding, BANG, this is negated: he CAN kill you and still escape, then kill again.
And the whole match speeds down the slope into one-sided kicking by a single champ.
And that's why Yasuo is hated: in SoloQ it's mostly enough to master early start to dominate the whole game. Once you stand against player who can't counter you... voila! You become a kill-them-all Pentakill-maker.
PS: What to do? Reduce their ability to kill champions fast, one by one. A good assassin should be ably to kill one, maybe two at the most. Then he should not participate in a battle anymore unless on very low HP under a constant threat of being killed. Kill one is okay: take down ADC then let your team do the rest. Or ADC and APC if lucky. But not all one by one!