When NASA calculates a trajectory of a space probe it uses the force of gravity defined by Newton. NASA does not use curvature of spacetime.
Attached is a description of how NASA calculates these slingshots to execute a change in a probe's trajectory; a video is provided also. NASA has certainly demonstrated their technique with numerous successful missions.
The calculation of a slingshot involves these critical values (and more):
a) the mass of the probe
b) the mass of the planet
c) the velocity of the probe
d) the velocity of the planet.
During the probe's approach there is the mutual force of gravity between the two bodies where the paths of both bodies are affected simultaneously. Obviously the probe with a rather small mass is affected much more than the planet.
These calculations are based on simple Newtonian mechanics.
Relativity is based on spacetime curvature by a gravitational field. A gravitational field provides free fall acceleration toward that body which is spherical having uniform density. Conforming to those rules this body exerts this field which can be calculated from the mass density and radius.
The mass of the observer, a smaller mass than the main body, is not involved in this free fall calculation.
On Earth, applying a force to a body to lift it gives the body potential energy. Upon releasing the body it will have free fall acceleration toward the main body.
Curvature transforms this gravitational field into a distortion of the observer's reference frame or just changes in the observer's coordinate system.
Curvature never involves the mass of the observer as it is not involved in a free fall acceleration.
Curvature also never describes an affect on the body exerting this gravitational field which is affecting the observer's path. Relativity is limited to only the observer and their reference frame.
NASA never uses a gravitational field in its calculations for a slingshot trajectory. NASA does not use spacetime curvature.
Relativity assumed gravity had a velocity limit of c. NASA assumes gravity is instantaneous.
While not a disproof of relativity this post just shows relativity's spacetime would not work for NASA and was never used.