Honestly, Illaoi's main problem is her incredibly low base Attack Speed
if she had a higher base Attack Speed she would have a much easier time in lane, but trying to trade and farm with a base AS of .571 when everyone else has .625 or higher is super brutal
she also runs out of mana super fast after a fight, but given how much damage she does I think thats fine.
Its weird that games of her where I run AS runes vs when I don't are drastically different. With AS runes (even if just 10-15%) its like night and day. You can actually farm minions and hit your spirit more than 3 times. Doesn't seem like it would matter much on paper but in reality its ludicrous how much it changes things.
I also honestly think even if that did that change, just like Mordekaiser, the majority of players don't build her properly which makes her really crappy. I have found that if you don't build Titanic Hydra immediately you don't really do anything. You need that item in order to trade, even if it seems like Cleaver and Sterak's are more important. (and in terms of Morde, Riot actually believes that hes fine currently at least according to Riot Meddler. Hint: follow Korean build)
Just the thoughts of a Silver Illaoi player.