You don't say what kind of a group $G$ is but I'm going to assume for simplicity that $G$ is finite. Then, yes, it follows from Artin-Wedderburn. The point is that once we know that $\mathbb{C}[G] \cong \prod \text{End}(V_i)$ as algebras we also know what modules look like in terms of the RHS: modules over a finite direct product canonically break up into a finite direct sum of modules over each factor, and modules over $\text{End}(V_i)$ are direct sums of copies of $V_i$. 

So if we consider the action of $\mathbb{C}[G]$ on a representation $V \cong \bigoplus_i n_i V_i$ then it follows immediately that the factors $\text{End}(V_j)$ where $n_j = 0$ act trivially, while the factors $\text{End}(V_i)$ where $n_i > 0$ act on each individual isotypic component and hence linear independently. 

It may be more conceptually satisfying to instead use here the <a href="https://en.wikipedia.org/wiki/Jacobson_density_theorem">Jacobson density theorem</a> or the <a href="https://qchu.wordpress.com/2012/11/11/the-double-commutant-theorem/">double commutant theorem</a>.