I agree. We also have the additional experimental fact that we have never observed anything that indicates an edge of spacetime with finite curvature. If the spacetime didn’t end then the geodesic could be further extended. Anyway, consider the standard KS coordinates covering the maximally extended Schwarzschild vacuum. The singularity in region II is an edge of the manifold. https://en.m.wikipedia.org/wiki/Kruskal–Szekeres_coordinates That option definitely is considered. That is one of the drivers of work on quantum gravity.