OP's question (v2) seems partially caused by imprecise use of the word local:

1. If OP's eq. (1) holds locally in a neighborhood $U\subseteq M$, then the (Levi-Civita) Riemann curvature tensor vanishes in $U$, and the manifold $M$ is by definition flat in $U$.

2. For an arbitrary point $p\in M$ on a Lorentzian manifold $(M,g)$, there exist Riemann normal coordinates in a coordinate neighborhood $U\subseteq M$ of the point $p$ such that the metric $g_{\mu\nu}$ becomes on Minkowski-form with vanishing (Levi-Civita) Christoffel symbols $\Gamma^{\lambda}_{\mu\nu}$ locally in the point $p$ (but not necessarily in the punctured neighborhood $U\backslash\{p\}$).

OP's question (v2) seems partially caused by imprecise use of the word local:

1. If OP's eq. (1) holds locally in a neighborhood $U\subseteq M$, then there exists coordinates in $U$ such that the metric $g_{\mu\nu}$ becomes on Minkowski-form in $U$, and then the (Levi-Civita) Riemann curvature tensor $R^{\sigma}{}_{\mu\nu\lambda}$ vanishes in $U$, or eqivalently, the manifold $M$ is by definition flat in $U$. The implications also hold in the opposite direction, after possibly going to a smaller neighborhood $V\subseteq U$.

2. For an arbitrary point $p\in M$ on a Lorentzian manifold $(M,g)$, there exist Riemann normal coordinates in a sufficiently small coordinate neighborhood $U\subseteq M$ of the point $p$ such that the metric $g_{\mu\nu}$ becomes on Minkowski-form with vanishing (Levi-Civita) Christoffel symbols $\Gamma^{\lambda}_{\mu\nu}$ locally in the point $p$ (but not necessarily in the punctured neighborhood $U\backslash\{p\}$ and the manifold $M$ is not necessarily flat in $U$). In particular, the (Levi-Civita) Riemann curvature tensor $R^{\sigma}{}_{\mu\nu\lambda}$ does not necessarily vanish at $p$.