You are correct. It's a fortunate coincidence and nothing more. It's much easier to think of the gradual warping of spacetime around a black hole so that more

You are correct. It's a fortunate coincidence and nothing more. It's much easier to think of the gradual warping of spacetime around a black hole so that more and more directions are down. East? No, that's really east and down. Left? Nope, you thought you were going left but it is left and down. Less and less "up" is available to you until you are on a knife edge where you can make an orbit (assuming a non-rotating black hole, no angular momentum) around the event horizon with a photon. Beyond that, everything is some flavor of down.