I honestly don't think he's that fun. I dunno if that helps your quest to find answers as to why Lee is a popular champion, but if it helps, these are the champions I do enjoy playing:
Both top and jungle. He's slippery, flexible, has a cool theme, and is difficult to play, but when played perfectly is difficult to beat. An average Shaco won't cut it against most junglers, but a perfect one will control both jungles.
Pretty fun to watch melee champions' futile chase while their health steadily falls, then Ahri turns on them. Also fun to catch out stray carries, or weave through teamfights to get to the squishies.
A badass undead centaur that speedily rushes through the jungle to dive into the enemy team? Yes please.
Torturous pirate king that blows things up and shoots cannons. Deceptively hard to kill with his movement speed buffs, slows, and cc removing heal.
Fast paced underdog that turns a losing fight into a winning one.
Badass prince that flags 'n' drags his way into their team, separating all from the unlucky soul that's about to hear the true champions' home town. DEMACIA
Deceptive assassin needs to wait until the perfect moment to jump onto that carry. Also very good at kiting, which I enjoy.
Awesome theme, aggressive in nature, plays unlike any ADC in this game.
Versatility incarnate. Also extremely good at kiting.
I compare her to Sheik from Super Smash Bros (project M specifically), you've got to whittle your opponent down and pick the perfect time to jump in and out of fights to maximize potential. Out of every assassin in this game, she genuinely feels like an assassin.
Zany explosive addict with flashy spells, what's not to love?
Tiny master of evil that disrupts and decimates all that stand in his way. Also very good at kiting.
Tanky sustain damage dealer that can chill in the middle of a teamfight dealing high damage and not have to worry about being blown up immediately.
A motherfuckin' dragon. Tons of aoe damage, insanely fast jungler clears (making jungle a lot less boring than usual), a solid champion no doubt.
The most mage-like of all ADC. I enjoy skillshots and he's got a bunch. Flexibility lets me play him mid or adc, ap or ad. Crazy good at kiting.