Honestly, the "assassin fantasy" in a game like league isn't really some kind of "rogue" or "thief" like you might think in other games. It more accurately fits the description of taking a fighter, bulling full damage, and "popping off".
Basically, the abstract definition of an "assassin" in league is fundamentally based on what you might expect from watching a youtube highlight, playing/watching a smurf of a high level player, or an entire class designed to play like a LCS playback. Rather than a layman definition of what an "assassin" might do.
Mind you, they're also the newest design in the game. They started to see actual development support around season 3 or so, mostly with the release of, I believe in this order, Rengar, Diana (she was released as an assassin), then Kha Zix. Prior to that you saw early attempts to create kind of mobile melee mage style champions like Fizz, but their mechanical advantage was generally seen as extremely toxic and unfun.
That said, particularly, AD assassins have been pushing the game into the current situation where everything is an insta kill. Supporting them in development created an AD arms race by treating AD as a kind of Ability Power was a huge mistake by riot, and will constantly cause problems until rectified.
Really I think assassins should be more like executioners, a specialized class designed to jump in and clean up low health enemies (of any class), but getting punished if they misjudge their burst damage and waste their CDs without managing a kill. Think resets, % missing HP damage, and decent farming tools. They could get CC, but I think that should also scale with % missing HP, and let's be honest, how terrifying would it be to be at 25% HP, then you get slapped with a 5 second grounded effect, as you desperately try to make it back to safety while being stalked by someone about to finish you off. < Now THAT is an assassin fantasy