There is no reason for this, and the answer is indeed no.

The simplest example I can think of is the product of two $\mathbb{S}^2$, each endowed with the round metric. This manifold is homogeneous and thus has constant scalar curvature, its sectional curvature is non-negative so its Ricci tensor also is (and is in fact even positive), but the Ricci curvature in a direction $u$ depends on the angle between $u$ and the tangent spaces to the fibers of the projection on each factor (i.e., on whether $u$ is close to be horizontal or vertical or not).

There is no reason for this, and the answer is indeed no.

The simplest example I can think of is the product of two $\mathbb{S}^2$, each endowed with round metrics of different radius (added in edit). This manifold is homogeneous and thus has constant scalar curvature, its sectional curvature is non-negative so its Ricci tensor also is (and is in fact even positive), but the Ricci curvature in a direction $u$ is not constant (directions tangent to the largest radius sphere have smallest Ricci curvature).