Her "AoE Stun" is only an aoe if A. your entire team is in a konga line, or B. she recently used her ultimate and used her E to shotgun it.
"you're dead post six if you get stunned" it's called "build MR" or "Play Akali/Fizz"
She lacks counterplay early on? She's completely vulnerable to ganks because she has no escapes and only one CC, two if you count an unreliable slow, she's very weak to certain champions like akali or fizz, and a ton of people can easily dodge or negate her stun. Morgana, Fizz, Akali, Kat, Yasuo, the list goes on.
I do agree that her early game damage should be toned down but that's mainly because her base damage is too high. Lowering her base by 10-25 on her Q would probably help balance her a little better. And this is coming from a Syndra main.