She was the human element. God, having created humans, could have done something without human agency I suppose, and still have become a human. But, He chose to use a woman. He did not cease being God in the interim between His enthronement in Heaven and his gestation in Mary's womb. Therefore, no. She didn't give birth to the divine person of Christ in the sense that she caused him to exist. Since God the Father, God the Son, and God the Holy Spirit are one, and God was still 100% God even at the moment of conception - no, I don't believe Mary gave birth to God since giving birth implies that God had a beginning. The human part of Jesus had a beginning. But even before conception, He was already God.
I know the church has never officially taught Mary as divine (though it has come mighty close with some of its doctrines- speaking of RC church here); but my point in calling her "Mother of God" is that there was absolutely NOTHING that Mary did to contribute to Jesus's divinity. Zilch. She contributed to his humanity, but not his divinity since she had no divinity in and of herself to contribute.
The misunderstanding of the phrase 'Mother of God' you are objecting to is one reason that we Orthodox tend to leave the technical Greek theological term in Greek even in your English translations of hymnograpy: Theotokos.
The infelicitous Englishing of the term as "Birthgiver of God" is closer to the sense of the original Greek than "Mother of God".
You really ought to read the decrees of the Third Ecumenical Council against Nestorius to get the full sense of the term, and why the title is necessary to a proper understanding of Christ.