Zed has tons of weaknesses 1 vs 1.
His 1-3 laning is so bad against any competent ranged attack opponent that he'll lose most/all of his potions right there before he even gets his full combo.
His laning is nowhere near as strong as it is commonly believed. He relies on his WEQ combo which is not the hardest thing to dodge. Every time he misses, you have a 20+ second window to punish him because his combo is down (if he just Qs it does next to no damage so long as its minion blocked). Also, keep in mind that psychologically you have an advantage over Zed in laning. You don't have to kill him, you'll simply have more utility than him late game. He on the other hand, MUST snowball or become useless. This can lead to a lot of stupid plays from him.
Late game he has a ton of damage but because he's skill shot based and generally has to get up close to score kills, he has the potential to mess up at any point in the game and is under ridiculous amounts of pressure. Not to mention all of the counter-itemization options available (kiting while taking Bloodthirster, Zhonyas on AP Mages, GA on ADCs etc).
Simply having a Support like Zilean or Taric is almost by itself enough to make his late game teamfighting capability irrelevant.