What's wrong with this proof that irreps over $\mathbb{C}$ are dimension $1$? Let $V$ be an irreducible representation of a finite group $G$ over the complex numbers $\mathbb{C}$.
Then $[V, V]_{\mathbb{C}[G]} \cong \mathbb{C}$ since $\mathbb{C}$ is algebraically closed. It follows that $\mathbb{C}[G]$ acts on $V$ by scalars, so that all $\mathbb{C}$-subspaces of $V$ are $G$-invariant. Therefore $V$ must have only $2$ subspaces as a $k$-vector space, so that it must be $1$-dimensional.
On the other hand, $S_3$ has an irreducible $2$ dimensional representation, $\mathbb{C}^2$, which occurs as a sub representation of the canonical representation of $S_3$ on $\mathbb{C}^3$.

I'm not sure why you'd say "it follows $\mathbb C[G]$ acts on $V$ by scalars."
Taking your example of $\mathbb C[S_3]$, you know it is a noncommutative semisimple ring, and you are aware that $M_2(\mathbb C)$ appears in its factoriation. The way $\mathbb C[S_3]$ acts on the simple submodule for that piece is not via a scalar, but via $M_2(\mathbb C)$.
