No Catholic has ever said that Mary is the origin of the eternal God.
Mary became the mother of Jesus Christ at the moment of his conception, in her womb.
Because Jesus Christ is God, Mary is called the Mother of God.
No one has ever said that Mary is in some way the origin or source of the eternal God or any one of the persons of the Trinity.
Saying that “I reject the notion that Mary is the Mother of God because Mary did not pre-exist God” is like saying, “I reject the notion that the earth is round because fire engines are red.”
Mary did not, of course, pre-exist God. Mary became the Mother of God at the moment she became the mother of Jesus, which occurred when she conceived him in her womb.
Thus the confusion that Rome has created.
Hail Mary; Mother of JESUS has EXACTLY the same 'meaning'; but it does NOT have the magical power that Mother of GOD has; does it.