As far as I know, the geodesic equation of motion can be directly derived from the equivalent principle. For instance, as shown by Steven Weinberg, the geodesic equation can be obtained by transforming a zero acceleration on the free-falling coordinate system to an arbitrary coordinate system.
Meanwhile, it seems that such a free-falling coordinate system can only be constructed under a very strong constraint. That is, the free-falling coordinate system can be defined and maintained, only when an observer and a target particle are at rest with respect to each other initially or at any time during their free-falling process. This is because an initial difference in velocity between the free-falling observer and particle will be increased over time in a gravitational field.
With regard to this issue, I would like to emphasize that the geodesic equation is used as an equation of motion itself, which is used to describe a motion of an object with an arbitrary initial condition, not as an auxiliary equation for determining a physical state of a system. To address this issue, is it necessary to add something else in the known geodesic equation?