In Siksek's notes The modular approach to Diophantine equations he uses the following result:

Let $p$ be an odd prime. For an elliptic curve $E$ over $\mathbb{Q}$ if $p|E(\mathbb{F}_{\ell})$ then for all but finitely many primes $\ell$ then from Chebotarev's Density Theorem we conclude that $E/\mathbb{Q}$ has a $p$ isogeny.

Since he uses Chebotarev's Density Theorem the proof certainly involves the Galois representation $\rho_{E,\ell}$ but I struggle to see how