As I think you admit you can interpret and define g_uv(x) any way you want (so long as you have a unique description of individual events). Who is going to stop you? You can even define your laboratory to describe events by a "curved space-time", but your definition does not now create gravity.
JS: No, this is actually wrong in the sense of being too broad - too mathematical without the constraints of physics.
The first function of guv and its Christoffel symbols is to describe measurements on single test particles using accelerating detectors. The test particles can be in arbitrary states of motion.
The second function is to describe measurement on PAIRs of closely spaced GEODESIC (non-accelerating) test particles to see if there is any frame-independent CURVATURE.
The mathematical freedom is much greater than the physical constraints. This is why Feynman warns of "rigor mortis."
All physics must be grounded in practical operational definitions (P.W. Bridgman), otherwise it's bad physics in my opinion.