“Mother of Christ” (or rather in Greek, Christotokos) was the title Nestorius (and before him Theodore of Mopsuestia) preferred. The problem with it can be seen on this thread: folks who (like Nestorius) object to Mary being titled Theotokos end up separating Christ’s divinity and humanity in some way which does not reflect the reality of the Son’s Incarnation, if not the way Nestorius did, then in some way even further from the truth.
The person whom Mary bore is fully God. She did not give birth to Jesus’s humanity only, but to the Divine Logos who in His conception according to the flesh has assumed our nature entirely (excepting sin). The title Theotokos safeguards the correct understanding of this mystery. It is not about Mary, but about Jesus.
Well, it’s become about Mary and the best thing to safeguard the correct understanding of Christ is good teaching.
Besides, it doesn’t affect Jesus if someone denies either His divinity or His humanity. He doesn’t need someone to stick up for Him. What it does make a difference to is the person who denies either aspect because then they aren’t believing in the Christ of Scripture and that false Jesus is not someone who can save them.
I know one thing, though, and that’s that many people come to Christ and are saved by Him without having all of their theology all hammered out ahead of time. Some people haven’t been raised with the theological teaching necessary to get an in depth understanding of who Jesus really is but they do know that He can save and He does.
Then the Holy Spirit will take care of the details.