Your first part was exactly correct by my understanding, gravity is not a force, it doesn't act on matter, so matter doesn't feel it, because in a over-simplification it essentially doesn't exist as far as matter is concerned.

Matter, the Earth, you and I, we occupy the three spatial dimensions of our universe where gravity does not act; we occupy the space part of spacetime. Space itself doesn't occupy time, but it is intertwined with it, and thus consequence is shared bidirectionally.

In my mind I treat it somewhat like a train, space is the carriage with things inside, and time is the track and locomotive. They are separate, but one can't be without the other (technically however in this analogy yes they can be, so it's not perfect).

Gravity can bend the track, which will result in space going in a different way, because it has to follow the locomotive (time).

Now, relative to inside the carriage (space), things will move, anything that is loose will likely drift around, especially if it's a tight bend; that's why gravity does manifest spatial forces such as weight, inertia, G-force, etc.

Orbits, say our moon orbiting us, are actually straight lines relative to the orbiting body; Earth doesn't "know" to curve around the sun, it has no "lateral" force, it is propagating through space in a straight line with velocity given to it billions of years ago. However, again, Earth is matter, which only occupies space, so while Earth is going along a straight line in space, inside that train carriage, spacetime, the locomotive and track, is bent into an ellipse due to gravity. Because matter travels in a straight line relative to itself that intrinsically resolves to a spacetime geodesic (again, to go back to the aircraft, they follow a straight line relative to themselves).

Again, not an expert, just a young adult with a fascination about space and time. :)