My conclusion on why Lee Sin is so hated (+other champs)
He's been extremely popular for a long period of time.
It's that simple.
WARNING: UNPOPULAR OPINION SECTION: There are, realistically, no issues with Lee's kit that are far above and beyond those of other champions. His mobility is not the best in the game. It's up there for sure, but he's not frickin' Leblanc with two unconditional dashes that also give 2 blinks, or Kass with 3+ low CD blinks or even Quinn/Rammus/Udyr with consistent high MS. He has a fairly narrow skillshot-based dash and a low-medium range targeted dash that requires either a wasted ward or an allied unit. His damage early-on is high, but that's his entire role in the early-game. His ult scaling is fantastic (200% bonus AD), but if he builds enough damage to take advantage of it he's incredibly squishy. Finally, he's really not that overpowered. Even in competitive he's usually only the second or third best jungler at any given time.
And even if you disagree with all of that, which I'm sure many of you will given how these Boards work, there's pretty much no denying he's on the same level of pure cancer that is Fizz or Rengar, or any of the other various hated champions we have.
No, people hate Lee because they see him too often. When a champion is picked over and over and over, in Lee's case for 3 years , they will get tired of the exact same gameplay, regardless of whether it is well designed or not. It also makes a champion's flaws, even if they only have a few (not that Lee only has a few, mind you), drastically more apparent. This does not just apply to Lee, it applies to pretty much every popular champion we've had for a long time: Yasuo, Ahri (ok seriously guys if Ahri was at a 5% pickrate nobody would even mention her), Zed, Riven, Thresh, etc. Need an example? Zed is probably the most hated assassin in the game, but is also pretty much objectively the healthiest. Why? You see him constantly. Relatively small flaws is his design, like his ranged farming with W-Q, have become absolutely reviled over time when really he's nowhere near as bad as his competition. It's completely illogical, but that's how it works.
So what can Riot do? In my opinion, make these champions more niche if possible. Straight nerfs aren't going to do crap, but actively discouraging a player from picking a champion every game would do wonders. Example: I main Xerath. Even though I main Xerath and love his gameplay, I am not stupid enough to first pick him if Fizz and LeBlanc are open, because chances are if one of those is picked I will lose lane and be a detriment to my team. Champions like Lee NEED poor or very poor matchups against some champions to discourage them from being common picks. Obviously I doubt this would work for all problem champions in this catagory; I doubt Ahri's playrate will drop until Riot visually reworks her into an ugly earth spirit hag with the personality of Hillary Clinton. But it'd be a start, and something for Riot to look at after the mage reworks.
Oh right, and, uh, remember to waste several thousand dollars play fantasy esports on AlphaDraft. Just use the code "BLINDMONK" to get 31.51% off your first season!
(note: value may change based on Lee Sin's current pick rate)

combine with the damage of a burst assassin
with de defense of a tank
... Guess what kid, neither does the rest of the champion pool.
you won't be bursting anyone, but that isn't your job.
: Great initiator, poor escape/engage and rely on his high CD to create a good engage.
: All in with no escape, also rely on CD.
: No escape, rely on CD.
: Rely on his R to initiate, no escape.
: Tanky, Semi-Escape with her dash, but rely on her R to engage.
: Have to walk all the way to initiate and escape, no burst.
: No escape, rely on his Q+Ult or Flash+Ult to create a pick or engage.
: **Sustain, 2 engages that can be used to escape, great base damage, absurd scaling, Ult is a disrupt heavy CC NUKE with 600 base damage + %Max HP + 200% scaling. **
not fall under the category of a poor matchup for

. Lee doesn't need gold, but scales well with gold, farms relatively fast and has great ganks. There aren't many champs that combine all 3 characteristics of the ideal jungler (item independent but can scale, farms fast, good ganks on low CD). Those who do
tend to be overwhelmingly picked in competitive.