Did you ever play the old Evelynn? The old Evelynn had Hate Spike hit every single time, as long as there was a target within proximity (It was like Morgana's R, in that it would gray out unless a target was in range)
The Q skill shot followed by 3 uses was a contingent way to dull that spammability down and to more or less keep the old skill, while punishing those who poorly use said skill shot, and bring each initial hit's damage down.
Keep in mind several things:
-Q only does half of its damage to the target that the skill shot hit. It doesn't do that damage to OTHER targets.
-It also has smaller scaling than her old Hate spike, with her damage mostly being moved into her E and R.
-If you miss the skill shot, you cannot spam the skill. It goes on an 8 second cooldown, which is usually a DEATH SENTENCE for Evelynn.
-The skill shot can be BLOCKED by other targets. Just because you W someone doesn't guarantee that you will get the opportunity to charm them. If the Q is blocked, the additional damage will proc on the secondary target, NOT the target you want it to, reducing Eve's champ damage.
-The followup Q's do NOT necessarily home in on the target you want them to! if you have 6 minions and a Champ, Q often chases the targets nearest Evelynn, not the champion. You have to set up a line of sight between the nearest target and the champion you want to hit, which often means...... Yes. the Q's per cast damage will be halfed because the initial hit will be to a MINION.
Evelynn isn't busted in terms of win rate.
She IS busted in terms of single skill damage with no mitigation available.
She has literally the ONLY point and click 1 shot in the game; Empowered E. You can't miss E; No one can dodge it (Except Jax and Pantheon). But if Evelynn hits you with it on 3-4 items, she will have enough AP to one shot you, because E is based on % of Max HP+ base damage. I've done literally 95% of a Darius with 5k HP's health in a single E. Given proper Luden's procs and an active W, that damage can 1 shot.
The only catch: she has to get close enough to use it (usually a GLP, or Flash, or using her Ult as a gap closer)
However; even though the skill can ensure a 1 shot, it doesn't have enough presence given her limited mobility to turn around most games. She'll usually go around the map killing people, while the rest of her team is 4v5ing lane pushes. So even though she has a "busted" skill, she's not generally busted overall.
(She's like a weaker Zed; spends the entire game wracking up kills, and hoping her team pushes lanes. When they don't Zed/Eve lose).