If I called Mary God's mother, I felt I was saying the human being born at a certain time in history (a creature) had "created," had given life to, God Himself, Who had not previously existed. That was hard for me.
As you now know, that was never the intent of the phrase or title in the first place. With Protestants who reason in this non-historical fashion (not knowing what the history of this was), it is always good to clarify: "mother of God the Son." This is what the phrase means anyway, and they can't argue with that logic. It immediately clarifies it and doesn't allow the misunderstanding to come up at all.
If they deny that she is the mother of God (the Son), then they deny that Jesus is God, which they don't want to do.This resolves the problem altogether. But she is not only the mother of Jesus' human nature, because motherhood is about giving birth to persons, not natures (or souls, as in our case, when mothers give birth).
If, on the other hand, they deny that she is the mother of God (the Son), then they deny the Virgin Birth, and in effect, also the Incarnation, which they don't want to do, either.
See related papers: