Instead of spacetime curvature, it is better to think of a different system of coordinates.
For example, if someone is at 1 km from the North Pole, all the land (or ice) until the horizon in all directions (a radius about 5 km) can be approximately by a flat surface.
Even being flat, during any path in straight line, except if it is radial to the pole, the route is not constant, if by route we understand a given direction ($221^\circ$ for example, when $0^\circ$ is direction North. It is a consequence of using polar coordinates. But when the calculations are corrected for the effect of that curvilinear coordinates (covariant derivative) they show a constant straight velocity.
In a similar way, in our daily experience of a constant $g$ (approximately flat spacetime), any falling objects has a non uniform velocity in our coordinates of space and time (and also according to our senses in this case). But it also follows a straight constant velocity, in the meaning that the covariant derivative of the velocity is zero.