The way God achieves His purposes is not the way that a man would think.  But God was not raising just a king for a nation, but He was raising a psalmist for the Kingdom of God, who wouldn't only sing worship songs but live a worshippers life.

David was not anointed because of his office but because of his heart.   God does not raise a man or a woman, but He raises a message. And the degree that God can anoint that person is to the degree that the message or word has become flesh in them.  God will anoint that person whose heart and life is consistent with the message he carries.