The curvature of spacetime is a property of the spacetime manifold itself, it is not related to any particular choice of coordinates. The very essence of relativity lies in the Einstein field equations, which, colloquially, tell you that the energy-matter content of spacetime determines its curvature and metric. There is no formulation of general relativity known to me that avoids the idea of curved spacetime.
But this is not worrying - generally, such a geometric theory of physics is the kind we want. For classical mechanics, such a geometric picture is given by the Hamiltonian formalism and its symplectic manifolds. For gauge theories describing the other fundamental forces except for gravity, the geometric picture is given by the theory of principal bundles and the (e.g. electromagnetic) physical fields are amenable to be described as the curvature of such bundles (and wonderfully analogous to formulating GR with jet and frame bundles). For gravity, the geometric picture is that it is spacetime itself that has curvature. We don't even want to get rid of a description in terms of curvature, generally speaking.
Just because you can locally choose frames such that, at least at one point, the metric is flat, and hence you have SR, this is different from saying that the curvature is not required - a spacetime is flat only when there is a choice of coordinates such that the metric is flat everywhere.