There isn’t really any kind of ‘action’ in the mechanistic sense of the word. It’s just that test particles and their world lines are themselves part of spacetime, so they cannot do anything other than follow its underlying geometry. There’s no duality of any kind. See below analogy for clarity.
As I’ve mentioned in my last post, there is information, in the form of the metric which determines the relationship between points. So it isn’t a ‘zero set’. This is true even very far from any sources - even spacetime without gravity has geometric structure that is different from that of Euclidean space. This is (eg) why you can’t accelerate to the speed of light - the fundamental reason for this is geometric, so geometry has real measurable consequences.
It’s exactly like the calculus you learned at school - the derivative of a function is defined at a single point, yet gives you information about the slope of the entire function. That’s because what it really does is tell you about the relationship between neighbouring points on the graph of the function - how it changes from point to point. If you’re given just the (local) derivative, plus boundary conditions, you can reconstruct the entire function, even though any one single point of the function is just an (x,y) pair.
To give an analogy (!!!) - suppose you have two people starting out on different points along the equator, and flying north simultaneously at a constant altitude. When they start out on the equator, let them be - say - 1000miles apart. What happens? The further north they get, the smaller the distance between them becomes. Eventually they’ll meet at the pole. Why? There is no detectable ‘action’ or force between the two planes. Each plane starts off at 90 degree angle from the equator (so their trajectories are initially parallel), and they always fly straight (there’s never any detectable change in direction from their initial trajectory). Yet they approach one another. That’s because they are both confined to the surface of the Earth, which is a sphere; so they must follow its intrinsic geometry. The metric governing this has real, detectable consequences.
There is no detectable information about this at any one point on the Earth’s surface. This is because the geometry concerns relationships between points, so what you do is take measurements of path lengths, areas, or angles. For example, you’ll find that the sum of the angles in a triangle on Earth’s surface is no longer exactly 180 degrees - it’s possible to directly measure this deviation. But you can’t do it at a single point, you need to measure across some distance. That’s because the effects of a non-flat metric are accumulative - mathematically, you integrate components of the metric to obtain path lengths. To put it differently, the metric defines an inner product of tangent vectors, so it’s a local object, but with global effects across the manifold.
Similar principles are true for curved spacetime as well. You can measure path lengths through spacetime pretty much directly (Shapiro delay, Pound-Rebka, gravitational wave detectors,...) and find that they differ from what you’d expect in a flat geometry. You can also directly measure angular distortions in the geometry, ie gyroscopic precessions, frame dragging etc. Gravitational light deflection is in effect a demonstration of the angle sum in a large triangle being different from 180 degrees close to a massive body. And so on.