Just to be provocative (in reality I agree with you that light cones are more fundamental objects than any spacelike hypersurface, and that light cone surfaces are physically meaningful at least in the local neighbourhood of p) I'll drop in some Bob Geroch: "[Special Relativity] emerges as just one more physical system. It consists, just like the others, of certain fields subject to a certain system of first-order, quasilinear partial differential equations. The causal cones of special relativity (which reflect the speed of light) have no special significant over the causal cones of any of the many other such systems in physics." <https://arxiv.org/abs/1005.1614>
(That democracy of causal cones paper is surprisingly relevant to some of the conversations you are running here tonight, particularly §2. I think you might especially enjoy the last paragraph on p. 6, notably its last two sentences starting "We note that, quite generally, C is a nonempty, open convex cone of tangent vectors...". In the context of our discussion, you might not enjoy the part starting at the bottom of p. 7 to the start of §3 :-).)
(That democracy of causal cones paper is surprisingly relevant to some of the conversations you are running here tonight, particularly §2. I think you might especially enjoy the last paragraph on p. 6, notably its last two sentences starting "We note that, quite generally, C is a nonempty, open convex cone of tangent vectors...". In the context of our discussion, you might not enjoy the part starting at the bottom of p. 7 to the start of §3 :-).)