I love Aurelion Sol and will run 
exhaust on him in mid just to screw over ganks and turn the tides around on melee/dive champs. He rarely has any bad to the point of unplayable matchups (EX: Like Kass vs. Heim where Kass just loses automatically) but you have to know how to position and what you're doing/when you can go in. I'd argue Talon/Kass would be a nightmare for him, but at least you can still hold your lane and are hard to shove into your own turret. The W harass is generally good even when against assassins.
On the plus side, 
Zhonyas doesn't stop your passive.
However, if you just blow your stun against a champ that can zoom onto you, you die. If you underestimate an opponent and they dive you when you can't quite finish them off, you die. If you don't know aurelion sol's limits or how to position exactly where you need to be, you die.
I'd argue...for Aurelion Sol, if you don't play stupid and keep up with the game (which isn't hard unless the other lanes are failing and people are constantly in your own lane), you can be extremely strong with empowered passive W. He can kill entire teams/focus multiple people at once with serious damage over a long time. Is the fighter behind you and the ADC somewhere in front of you? NO PROBLEM! You should be building Rylai's and Rod of Ages, which should give you enough bulk to survive a teamfight.
Surprisingly, people underestimate Sol and are stupid enough to not focus you, because they don't realize how much that damage stacks up. If you are fighting a team that has hard CC/strong dive and actually knows to use it on you, then it gets tough...but that's what zhonya's is for. It doesn't stop your passive, so if people dive you, you can Zhonyas at the last moment and still be outputting basically the same damage as before while your ADC melts people when the fighters/assassins are thrown off for a second and have to switch targets.
The E is decent for pushing up waves, such as if you have baron buff or want to sneak an inhibitor. You push the wave up, start weakly hitting the turret, and as soon as you see somebody you E away (preferably through walls).
Is a 
Yasuo coming at you and you aren't dumb enough to stand in a minion wave? NO PROBLEM! Q-Stun/Exaust/Rylais him and then watch as he slowly kills himself trying to reach you.
This leaves Aurelion sol with a lot of tools/potential if played intelligently:
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There are very few lanes he is "useless" in, unless you're playing him poorly. Even then, he can normally punish or even kill his counters in the early levels with well placed Ws and Exhausts/Qs. (Melee-range AA champs trying to farm, in particular.) When dived, if Aurelion Sol survives with just enough health, you can go and flash out, turn around with a Q, and then kill the enemy with Ws if you're skilled enough.
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Can shove in lane pretty easily thanks to his W in order to roam, and can return quickly to lane with his E. Otherwise, using his E to gank makes his ganks almost unwardable. His ganks are also strong, thanks to his Q.
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Can completely defend himself from those dive/melee/CC weaknesses in a teamfight while still doing 80% of his damage with itemization.
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Can somewhat peel for his teammates. Q-stun stops pursuits and saves lives. Rylais+W helps as well.
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Can check bushes without using wards/his face by using his W and see if it hits anything in the bush.
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Can kill multiple people at once if positioned properly, since his passive is a large range in a circle. This means...yes, he can carry teamfights and even picks/games if he gets slightly ahead.
The biggest problems I've had with Aurelion Sol
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If the rest of the team does poorly during laning phase, he can't carry. He needs 2-3 items to really become a monster that trolls everyone via timed stuns and clever positioning.
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Skillshot stuns (EX: Taric, Lux, Morgana) are horrific for him. Because the stuns are done at range, he struggles to "return the favor" like he can with melee champs trying to apply CC/burst (See first point in "Aurelion's potential").
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Projectile/Magic shields are hard to deal with. This means Morgana is kind of a nightmare. A smart Yasuo will time his windwall well to get Aurelion Sol's stun down, then turn on Aurelion and kill him (Unless Aurelion has both exhaust and flash up to blow for an escape). Aurelion is very reliant on his stun to not die in certain matchups, and so it only takes that one window of weakness to semi-slowly kill him while he runs away, helplessly.
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His E isn't as useful as it could be. It's good for re-positioning/rejoining with your group or to get to lane, but it can't work as an escape tool unless you are careful. Relying on it to push a wave and then zoom out is risky and I've died a little too much trying to make it work - you need a team that has GOOD vision of the map to utilize it or you'll run into some sort of potential nightmare situation. E passive is pretty great though, as you can escape from the skin of your teeth with that movement speed buff. It just takes a little too long to come up for a reliable tool to get out of bad situations, even if you see them coming a mile away. He can't even really sneak inhibitors with it depending on where they are, because his passive procs monster aggro and it's hard to escape once you go in, making it essentially a suicide mission. (Though depending on who comes to deal with you, you might be able to kill them anyway and make it a good trade.)
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I feel he is extremely reliant on specific itemization for balance, Rylai's especially. If riot does much to that item, especially the item's slow passive, Aurelion Sol could easily become underpowered. On the flip side, if they introduce another suitable item or buff another item, he could become very powerful. After Rod of Ages/Rylai's/Boots and situational defense, he can kind of get whatever, so if another mage item comes into the game that just "fits" him it could get out of hand.
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He can push waves up to turrets easily but sucks at actually killing the turrets. This means relying on your team to get towers after you push them up; actually pushing up to tower can only be used to roam in the laning phase or distract during group phases. He can at least reasonably finish turrets off, however.
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Mana on Aurelion Sol is reasonable. But if your team drags out a fight/chase, you run out of mana sooner rather than later. You NEED W to do any real damage and you can't burst people. So, if your team is being sloppy, you can become useless and run around behind everyone trying to get just enough mana to land another Q so that you aren't useless despite being full health.
Why I think people don't enjoy playing Aurelion Sol
I love him, so this is more of a guess. But his playstyle is very unique and messing up with him normally gets you punished VERY HARD. His playstyle is very "careful" and somewhat deliberate.
*You can go pretty ballsy with him, but it means being sneaky/deceptive about how much damage you can actually do/can take, and requires basically ensnaring people in the exact right range to slowly stun and hit them with stars to death. For me, this is satisfying, because it's like seeing a trap play off or being able to turn a situation on people is very fun. But for others, this might feel too slow. But it also requires careful mechanics, vs. divers who just run in, spam combos/attack buttons that are on low cooldown, and run out. With Aurelion Sol, you have 5-10 second cooldowns for your 2 main damage abilities and you have Mana to worry about.
*To add to the first point: his kit feels more like playing an immobile ADC than a mage, and he doesn't get the brutal satisfaction of facemelting people like an ADC does. He melts multiple people at once and entire teams, but it is in chunks at a time. ADC: Right click to quickly melt one person. SOL: W to kind of nominally melt multiple people.
*It is probably super unfun to people when feel "helpless" to his counters. Feeling completely useless as a melee chases you down while still living long enough to give commands to Sol, ala running around helplessly and throwing Q-stuns that can't possibly land on the people next to him...sucks. (He gets bulky if built correctly, so helpless and painful deaths can last several seconds.) Either Aurelion Sol is chasing people down while they are helpless (or they don't realize what is happening to them, such as an ADC in a teamfight getting hit by the strong passive) OR...Aurelion Sol is being chased down while he is helpless.
*He doesn't feel "fast" or "bursty." I saw somebody compare him to an old, reliable diesel engine on the boards once, which I find accurate. This means there isn't an instant gratification for doing well with him for most people, though I enjoy setting people up to their doom. :)
Conclusion
He's just not going to appeal to everyone/the typical player [sneaky/deceptive, slow-burning, positioning reliant playstyle], and it doesn't help that people keep being snide/worry about how he is "weak" and banning him in champ select when their teammates try to pick him. (He's not bad or in a bad spot, I promise. He's kind of the AP version of illaoi.)
I think he might actually feel better for people who don't normally play mid/jungle, because he mirrors other playstyles (CC-spam, Peel, ADC) more than his fellow midlaners.
Ultimately I find him strong and self-sustaining with enough team-reliance and weaknesses to define him as balanced with a unique identity. His strengths are very strong and are worth his weaknesses unless you picked him into a nightmare team comp (ranged stuns + a Yasuo on the team).
had a chance to shine back in the day, Aurelion never got that chance. Anybody have any ideas on what we can do with Aurelion sol? He needs something, I just don't know what.
ADC: 
