At first glance, it seems the label of Mary as the "Mother of God" is blasphemous, since God is a Trinity of Father, Son and Holy Spirit. But the Gospel tells us that the God being referred to here, in which Mary is a mother of, is the second person of the Trinity, Jesus and not the whole trinity. Knowing that Jesus is God, we can rightly label Mary as the "Mother of God" being conscious that the God here is the second person of the Trinity.
So we have two choices:
1. Declaring Mary as "Mother of God" being conscious that the God here is the second person of the Trinity; or
2. Declaring Mary as "Mother of Jesus" being conscious that Jesus is God and is the second person of the Trinity.
The first option serves *so many* purposes than the second, in terms of preaching the Gospel.
First, it makes the mystery of Incarnation (the central mystery of Christianity, aside from the Trinity) so fully alive. Anyone who has not encountered Christianity hearing the phrase "Mother of God" will be compelled to ask: How can a God, an eternal Being, has a mother? Then we can answer him: God's love is tremendous that He came down to us, as a baby born in Bethlehem, to redeem us of our sins and reunite us again with the Triune God, the thing Adam and Eve lost in the Fall. So the three words "Mother of God" captures this mystery of incarnation. The Word made flesh.
Second, the phrase automatically profess the divinity of Christ, and removes the early heresy saying that Jesus is only human, not divine. This heresy is still alive today (e.g. Jehova's Witnesses), and hence the Church has been battling this heresy for a very long time. Proponents of this heresy even uses the Bible to prove that Jesus is only human. (This is one of the flaws of denying the living Magisterium and just subscribe to private interpretation of the Bible). So in declaring Mary as "Mother of God", we are putting forward Christ's divinity, not just a required awareness.
Third, officially calling Mary as the "Mother of God" separates us from these "christians" who deny Christ's divinity, since all of them (e.g. Jehova's Witnesses) declares Mary as a mother of Jesus, but without the additional conviction that Jesus is God.
Fourth, the title prevents the error of Nestorianism, the teaching that Christ's humanity and divinity are disunited. This heresy gives us two Jesuses: one human and one divine. The error is that we have two beings: one is created, and one is uncreated. But Jesus is only one, and he is 100% human and 100% divine. We cannot say that Mary gave birth to only the human Jesus because there is only one Jesus, and this includes his divinity. At some point, before the Second Person of the Trinity is born, the Second Person of the Trinity is purely spirit. After He is born, then he is now 100% human and 100% divine. So in a way, choosing Mary as "Mother of Jesus" with the attached reason that Mary gave birth to the human Jesus, then one has fallen to the error of Nestorianism.
All these reasons of choosing Mary as "Mother of God" is really Christ-centered. We don't call Mary as "Mother of God" as if she has supernatural powers above the Triune God. I don't see anything wrong calling Mary as "Mother of God" with the awareness that this God is Jesus.
We call Mary as the "Mother of God" because she gave birth to Jesus, our Lord, Saviour and ultimately, God.