Now, instead of demanding that the relevant triangle inequalities hold, as he does... don't demand that. Then configurations where the inequality is violated would correspond to curvature.

Nope... dS^2 between me and my future self 2 seconds from now is 2 seconds squared. But both have zero dS^2 with a sphere one light second away. No curvature is present.

If on the other hand you're talking about just the space part, then not even curvature can do what you're talking about.