Essentially there are two meanings of "geometric".This I don't understand. Of course, you can only measure gauge-invariant properties, but I think, we are really drifting too much apart from the original topic of this thread.
1) Gauge-invariant. Everyone agrees on this aspect of geometry.
2) Euclidean geometry is a model of measurements using straight edge, ruler, compass etc - external rigid instruments. When you say that a proper reference frame is physically realized, the physical realization is an external rigid apparatus, since it does not contribute to spacetime curvature. So in the sense of needing an external rigid apparatus, your view is more "geometric", not less.