@UriBader There still seems to be a jump required there. A priori, the image of $G'$ in $\mathrm{SL}_2(\mathbb F_\ell)$ is a possibly proper subgroup of $G_1\cap \mathrm{SL}_2(\mathbb F_\ell)$. So even if you can prove the result with $\mathrm{SL}_2$ more is still needed to prove the result for $\mathrm{GL}_2$. That seems to be Serre's approach (page iv-23 of Abelian l-adic representations and elliptic curves - Lemmas 2,3,5), but I was hoping there was a way of making sense of the above statement.

@A.Pacetti I'm specifically asking about CM forms.