Your mistake is that taking the Mostoski collapse does not preserve elementarity.

We do have a countable transitive $A$ and a countable $M$ with $$A\cong M\preccurlyeq V_{\omega_1},$$ where $M$ comes from downward Lowenheim-Skolem applied to $V_{\omega_1}$ and $A$ is the collapse of $M$, but that does not imply $A\preccurlyeq V_{\omega_1}$. Elementary substructurehood is more than just agreement of theories, it also takes into account exactly how the smaller structure sits inside the larger one. Moving the smaller structure around (e.g. via the Mostowski collapse) may break elementarity.

Your mistake is that taking the Mostowski collapse does not preserve elementarity.

We do have a countable transitive $A$ and a countable $M$ with $$A\cong M\preccurlyeq V_{\omega_1},$$ where $M$ comes from downward Lowenheim-Skolem applied to $V_{\omega_1}$ and $A$ is the collapse of $M$, but that does not imply $A\preccurlyeq V_{\omega_1}$. Elementary substructurehood is more than just agreement of theories, it also takes into account exactly how the smaller structure sits inside the larger one. Moving the smaller structure around (e.g. via the Mostowski collapse) may break elementarity.