Since the rotation values of the pivot game object are driven by a script, it means the animation stream must be updated in order to read from these new values when the pivot game object is nested in the animated hierarchy. Otherwise, the constraints will always read the default values unless that game object is animated by a clip or another constraint.
To fix your issue, simply add a `RigTransform` component to your pivot game object. This way, scene transforms will be read and fed through the stream for constraints to evaluate.
https://docs.unity3d.com/Packages/com.unity.animation.rigging@1.3/manual/RiggingWorkflow.html#rig-transform