I would appreciate if you could please evaluate my proof and point out any mistakes I made.
Proof:
Define a homomorphism $\Phi: \text{Aut}(G)\to \text{Out}(G)$, such that all elements of $\text{Out}(G) \leq \text{Aut}(G)$ are sent to $\text{Out}(G)$ (identically), and all elements of $\text{Inn}(G) \leq \text{Aut}(G)$ are sent to $1$. Thus $\Phi$ is surjective. Now let $\phi_a$ be an element of $\text{Inn}(G)$. Then $\phi_a(g) = aga^{-1}=aa^{-1}g=g \implies \phi_a = id_G \implies \text{Inn}(G)=\{id_G\}$. This implies that $\ker(\Phi)$ is empty, and hence $\Phi$ is injective. Therefore, $\Phi$ is an isomoprhism $\implies$ $\text{Aut}(G)\cong \text{Aut}(G)$.
Your proof does not seem to be true because as mentioned in comments $Out(G)$ is not even a subset of $Aut(G)$ in general. On the other hand, the proof of your result is very obvious. Indeed $Out(G)$ is just defined to be $Aut(G) / Inn(G)$. You proved that $Inn(G) = 1$. So we have $Aut(G) = Out(G)$.