Every act of Christ is education. He IS pure act. He IS the Word made flesh.
Why THAT psalm? Why only THOSE words? He could certainly have uttered any of the other words of Psalm 22 before expiring, but His "forsaken"-ness is front-and-center.
Since nothing Christ uttered was frivolous, how can you pass that off as some random snippet of a larger discussion Jesus was trying to have before dying (and cutting off that teaching moment?). Jesus spoke exactly what He willed to speak. We can infer a teaching moment in what comes after these words, but we cannot trivialize what He did choose to say - "My God, My God, why have You forsaken Me?"
Jesus was reciting prophesy showing it come to pass. He was not feeling as if he was forsaken and crying out.