Is it known why the sphere design was rejected, or was it simply not the right shape for the era when saucers were seen as the peak of space traveling?
I don't think it was anything as on-the-nose as that. Pato Guzman and Matt Jefferies went through dozens of design sketches with Roddenberry supervising, and the version they settled on was just the one that looked best to them. Aesthetically alone, regardless of any social trends at the time, the saucer design looks sleeker and more dynamic than a sphere. It also probably worked better as a model in terms of lighting and camera angles, because it looks different from different angles in a way that a sphere doesn't.