I mean all assassins live on a power budget. Some have CC, some have hyper mobility, some have sustain, some have stealth. Zed's power budget is pure damage and mobility on cooldown. Ekko has disengage, CC, slows, sustain. Akali has sustain, hyper stick potential, stealth. Rengar has stealth, tankiness, CC. I could go on. But Zed is just damage. Most assassins have had a phase where they were abused as tanks or alternative builds due to a problem with their kit. However, Zed has always been Zed. His kit does one thing and only one thing. And that's why Riot loves him. He's the perfect example of an assassin. Get in, get out, kill.
He's bitched about, but the link still highlights his counterplay. He's always been one of the most predictable and counterplayable assassins in League. The problem is that he can be a lane pub stomp. He defeats most conventional champions 1v1, and with enough mechanics and forethought, can beat just about any champion 1v1. Solo queue is a game of pride and the desire to be the best, and having a champion that can make you look like an idiot is infuriating, but his numbers don't show anything significant besides that he's disliked. Once you dissect why he's hated, it's not a balance issue so much as a general gameplay problem.
Your problem of him roaming bot is something all assassins share. Talon/Kassadin do it better. If people hate his mobility, Leblanc/Talon/Kassadin do it better. If people hate his CC, Ekko/Rengar/any assassin with CC does it better (Basically all of them but Katarina). If his burst is hated, Katarina/Talon/Rengar do it better. He's an assassin that is completely telegraphed. He can't do anything you can't see right on your screen. From where he pops out from his ult to seeing his shadows on the field. This is unlike Akali that shrouds or Talon that stealths and could just be liable to do anything in the blink of an eye. Katarina can shunpo to anyone/dagger in an instant without reaction time. He requires high coordination to play at a high level and has so much counterplay in a team setting.
And yes, I refer to boards complaining as bitching and crying because Boards is a tiny percent of the game's population, and is predominantly Bronze-silver players. Most people, including me, have no place talking about balance. The fact is if Zed was a serious problem, we'd see him in the LCK or consistently in high challenger. He's not.