$\DeclareMathOperator\SU{SU}$I am wondering if there is any known proof of the McKay correspondence (I will give the precise statement that I mean by this) that doesn't use the classification of Dynkin quivers or of finite subgroups of $\SU(2)$. Explicitly, I want to prove, without using the classification of Dynkin quivers or the classification of finite subgroups of $\SU(2)$, that
For $G \subset \SU(2)$ finite, the McKay quiver of $G$ is a Dynkin quiver.
Where we define the McKay quiver as the quiver given by (nontrivial) irreps of $G$ with an arrow from $V \to W$ if $V$is a summand of $\mathbb C^2 \otimes W$ (where $\mathbb C^2$ carries the natural action of $G$ as a subgroup of $\SU(2)$).
I know that this can be done by using the machinery of Bridgeland-King-Reid, but I am hoping that someone knows a simple/conceptual proof of this fact that can be taught to students. I feel that pedagogically, proofs relying on classification results can feel ``coincidental" in a way, and I hope there is a nice proof not relying on said classification results.
