Thanks for the answer! Do you know if there are any conditions that could be imposed to make this true? For example, in the case that I'm trying to apply this to, I have some information about the Lie algebra of $G$ -- in particular, I know that it's non-trivial.

@LSpice I mean in the natural representation $G(k)\subset GL(n, k)$

@YCor distinct means pairwise distinct. Specifically, I want to show that if $V$ is the closed subvariety of elements with a repeated eigenvalue, then $V$ has dimension strictly smaller than the dimension of $V$. I therefore need to rule out the case that $V$ contains a connected component of $G$.

(Maybe replacing Dirichlet density with natural density to avoid pathological sets of primes)

Thanks for your answer! Expecting an open subgroup was way too optimistic, but I guess the follow up question is do I know my representation on an open set? So does there exist a Galois number field $K$ and a conjugacy class $C\subset\mathrm{Gal}(K/\mathbb Q)$ such that $\{\mathrm{Frop}_p\} = C\implies p\in S$. In your example, $\mathbb Q(i)$ with the non-trivial automorphism would work.

@znt I should've said in the question - this does require $\ell>3$. Even if you intersect everything with $\mathrm{SL}_2$ you still run into the same problem trying to deduce the result for $\mathrm{GL}_2$. The fact that $\mathrm{SL}_2(\mathbb Z/\ell^n)$ is a perfect group does solve the problem, but the five lemma and perfect groups seems a lot to cram into a statement like "the result will follow".

@znt Any chance you can clarify? $G_n\cap\mathrm{SL}_2(\mathbb Z/\ell^n)$ is a subgroup of $\mathrm{SL}_2(\mathbb Z/\ell^n)$ containing $H_n$ and so is a union of $\mathrm{SL}_2(\mathbb Z/\ell^n)$-cosets of $H_n$. How does the induction hypothesis tell you that it contains an element of each coset? Have I misunderstood?