Local isometries preserve geodesics?

This is just an elaboration of the comment of @JohnMa . In fact all the concepts involved in you question are local, so they are compatible with restricting to an open subset. (You can restrict the metric and the Levi Civita connection and the restriction of the connection is the Levi-Civita connection of the restricted metric. This follows from uniqueness of the Levi-Civita connection, since the restricted connection is metric and torsion-free. This also is the argument needed for the missing step in your proof. Then you get that connections of the initial metric are connections for the restricted metric and so on.) Hence local isometries map geodesics to geodesics.