Why Ekko is unbalanced: an empirical perspective.
WARNING: This post is largely an exercise for my own amusement to quantify Ekko's unbalance. I have solid reasoning to back it up, a method to my deductions and 4 years of experience with this game as a foundation for this short bit of study. If you don't feel like slogging through a bit of a process, as well as a decent amount of math (although it is very basic math), please turn back here.
I see a lot of "Ekko is unbalanced" posts with very little, if any, effort to put a finger on WHAT exactly makes his kit broken. Everybody can tell, instinctively, that Ekko is a broken champion. They just KNOW he does too much damage, has too much in his kit. Well... they're right. But I decided to take an empirical approach to the subject and run out a little math.
As a control I decided to use Veigar. I actually think Veigar is the best comparable champion to Ekko, damage wise. If we were to look at his gameplay or utility I'd definitely rather compare to Akali, but utility is hard (if it's even possible) to exactly quantify, hence: Veigar.
Note: I'm making some assumptions in this analysis. That screws with the true objectivity of the math, but it would have literally been impossible to do it at all without them. If you have a suggestion to delete/replace any given assumption I'd be happy to take a look at the reasoning and plug it in, or even create an addendum at the end of this post.
My assumptions:
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The fight in which the two champions are battling lasts exactly ten seconds. This allows for me to account for multiple uses of a given ability, without adding in the variable of CDR. Both champions have strong reason to buy that particular stat, but it's still going to be up to player preference of whether or not they get any. So I'm keeping it clean, using base cooldowns and a long fight.
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Their fight is a full on 5v5 team fight. Even ground, no one caught out, no clear cut ability for any picks or wombo combos, just an all in, knock down, drag out brawl.
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Both champions are fighting enemies with 0 magic resist. We can assume they will be fighting similar enemies, aiming in particular for the backline and other squishies, so imagining a null MR scenario makes sense, and makes the math easier.
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Neither champion has Luden's Echo or Lich Bane. I know, this is a highly unlikely scenario, but both champions are just as likely to buy them (with Ekko probably being more likely to have both), but it's another item variable like the CDR. It's better to just assume that they both have neither.
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Both Champions are level 18, have enough mana to use their abilities every time they're available in the ten second fight, and live at least long enough to use their abilities the number of times ten seconds allows.
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Veigar is not attempting to auto attack at all, Ekko is not attempting to auto attack more than absolutely necessary to proc Resonance and do a little extra with the passive on Parallel Convergence
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Speaking of Parallel Convergence, I assumed Ekko managed to make use of the damage four(4) times throughout the fight, against an enemy with 2000 hp, who was missing exactly 1400 hp, leaving them at precisely 30%, before being attacked by Ekko.
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Similarly, I assumed Ekko managed to proc his Passive Z-Drive Resonance six(6) times.
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I also made a few assumptions on the number of enemies hit by a given Area of Effect attack. For example: I assumed Dark Matter had 60% efficacy both times it was used in the fight, for a total of six(6) instances of damage.
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I assume between Baleful Strike's kills, farm, and Deathcap's modifier, Veigar has 200 more AP than Ekko, this will be clearly taken into account in the math, with Veigar having 800 and Ekko 600 Ability Power.
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I assume Veigar attacks another mage with Primordial Burst, a mage with the same AP as Ekko (600)
Enough of that, on with the math!
Format: Ability - {[Ability Base Damage]+[Ability Power(AP Ratio)]}(Number of Casts per 10 seconds, if more than one)(Number of targets hit per cast, if applicable) = Ability Subtotal

Z-Drive Resonance(Passive) - {[231]+[600(.7)]}(6) = 3906 Timewinder(Q) - {[120]+[600(.2)]}(2)(2)+{[160]+[600(.6)]}(2)(2) = 3040 Parallel Convergence(W) - {[1400(.05)]+[1400(.0181)(600/100)]}(4) = 888.16 = 888 Phase Dive(E) - {[170]+[600(.2)]}(2) = 580 Chronobreak(R) - {[500]+[600(1.3)]}(2) = 2560
Total damage in 10 seconds: 10974

Equilibrium(Passive) - Non-Damaging Ability Baleful Strike(Q) - {[260]+[800(.6)]}(2) = 1480 Dark Matter(W) - {[320]+[800(1.0)]}(2)(3) = 6720 Event Horizon(E) - Non-Damaging Ability Primordial Burst(R) - {[500]+[800(1.0)]+[600(.8)]} = 1780
Total damage in 10 seconds: 9980
Final Tally: Ekko - 10974 | Veigar - 9980
Concluding Thoughts
LoL is part of the real world, it isn't, and cannot be, perfect. It will never be perfectly quantifiable with absolute precision.
But, I think I put some pretty thorough effort into my assumptions, and after typing this all out again, I think I may have even been a bit lenient on Veigar. Though I could just as easily have been lenient on Ekko.
I could start making grand sweeping statements about how this clearly proves Ekko's unbalance, but all I can do is my best estimation, and let the math do the talking.
Hopefully even if my post doesn't lead to direct discussion, someone may notice it and use it to further the discussion so that we can continue to help Riot improve the game we love.
Again, if anyone has any suggestions to change the formulae I used, I'd be happy to listen to them, and the reasoning behind them.