A consequence of the Tarski-Vaught test is the following: if $(M_i)$ is a chain of elementary end-extension, that is if: $$ M_0 \prec M_1 \prec \ldots \prec M_i \prec \ldots $$ Then for all $j$, we have: $$ M_j \prec \bigcup_i M_i $$
Then if we consider $L_{\omega_1}$ in the constructible universe, we can construct $(L_{\alpha_\nu})_{\nu}$ a chain of elementary substructures of $L_{\omega_1}$ by repeatedly applying the Löwenheim-Skolem Theorem. In particular, this implies that: $$ L_{\omega_1} \models \exists \alpha \, \forall \beta \, \exists \alpha' > \beta \, L_\alpha \prec L_{\alpha'} $$
Now consider $\alpha_0$ the least ordinal such that $L_{\alpha_0} \prec L_{\omega_1}$. The previous sentence can be reflected down to $L_{\alpha_0}$ and this implies the existence of some chain of elementary substructures $(L_{\beta_i})$ such that the $\beta_i$ are cofinal in $\alpha$. Hence, the previous consequence of the Tarski-Vaught test yields: $$ L_{\beta_0} \prec \bigcup_i L_{\beta_i} = L_{\alpha_0} \prec L_{\omega_1} $$ which contradict the minimality of $\alpha_0$. Where is the error in this reasonning?
There are several issues. It is true that there are arbitrarily large $\alpha$ below $\omega_1$ such that $L_\alpha\prec L_{\omega_1}$. That's different from saying $L_{\omega_1}\models\exists\alpha\forall\beta\exists\alpha'>\beta\ L_\alpha\prec L_{\alpha'}$, which says a totally different thing; although I feel like $L_{\omega_1}$ consistently satisfies this sentence, using something about zero sharp, it's probably not true in general, and moreover $\alpha<\alpha'<\alpha''$, $L_\alpha\prec L_{\alpha'}$ and $L_\alpha\prec L_{\alpha''}$ don't imply $L_{\alpha'}\prec L_{\alpha''}$, so you don't really get a chain.
The main issue is that "there are arbitrarily large $\alpha$ below $\gamma$ such that $L_\alpha\prec L_{\gamma}$" cannot be expressed as a statement $\varphi$ about $L_\gamma$, by your reflection argument; namely, there is no formula $\varphi$ such that $L_\gamma\models\varphi$ iff there are arbitrarily large $\alpha$ below $\gamma$ such that $L_\alpha\prec L_{\gamma}$ (or even "there exists some $\alpha$").