As grant mentioned, "Whether you can say that spacetime curvature is what really causes the deviation from straight-line movement is a philosophical point that I think has been debated on BAUT quite recently."

I do not believe gravity is a force at all, as F=ma, and yet gravity works in a finite manner on massless particles such as photons, thereby creating an apparent discontinuity via indeterminency. Yet GR's equations describe the effects on both particles with mass, as well as massless particles, with aplomb. We know spacetime curvature is real, the equations work, so I propose we stick with that framework instead of working out conceptual ways of shoehorning Newtonian mechanics into the a relativistic framework.