Christ did not consider himself forsaken nor was he pleading or begging for his life. He was reciting scripture on the cross. He was educating. He was holding court. He was providing his last sermon.
Was he forsaken?
I kind of see this a little differently. The verses in Psalm 22 were prophecies about the Messiah. Jesus fulfilled those prophecies. For example, how could he have known the soldiers would not break his legs which was SOP for crucifixions? Death from that cruel method was usually from suffocation but only after they suffered for hours. Breaking the legs prevented the condemned from lifting up to inhale. He was already dead when they came to do this. He didn't ask them to gamble for his clothes either.
Like I said, this passage was one of hundreds of Messianic prophecies. And Jesus fulfilled them all because he IS the Messiah!
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?"