This whole thread seems to be "Much Ado About Nothing." Whether the title is appropriate depends entirely on what one means by "Mother of God".
A basic tennent of Christianity is that Christ was 100% God and 100% human, a seeming paradox we refer to as the "Hypostatic Union". As far as I know, all major Christian Churches adhere to this concept. We also all agree that it is a mystery that defies true comprehension by humans, at least in this life. We can
state it, but we can't really understand it. There's more than adequate scriptural support for the concept, so "Sola Scriptura" Protestants defend the concept as vigoroursly as the EOC or RCC would.
Now, as Mary was the instrument of God in bringing forth the fully Divine Christ into the world, she was certainly the "Mother of God" at least in as far as that it was through her that God brought forth His Incarnate form into this world. She was a willing vessel, furthermore, and it was because of her yieldedness to God that she was chosen for this honor. If this is what one means by "Mother of God", then very few would consider the title inappropriate. She was the "Mother of The Incarnate Diety", and there's no two ways about that. If that's what you mean when you say, "Mother of God", then I can't imagine anyone having a problem with it.
On the other end of the spectrum, even the Pope himself would not claim that she in some way brought forth the "Elohim" of Genesis 1 that created the Heavnes and the Earth. That concept he would consider blasephemy as surely as would any Protestant. If that were what someone meant by "Mother of God" (and I've never even heard of anyone who means this), then surely that use of the title is entirely inappropriate, and frankly blasphemous.
Now, in between those two extremes are all sorts of shades of meaning, some of which might be appropriate, and some of which are not. But as the "Hypostatic Union" is itself a mystery, then the greyer shades of what it means to be "The Birth-Giver/Mother of God" are a mystery as well. Because we can't truly understand the union of human and Divine in Christ, we cannot fully understand everything about Mary being His Mother. I don't know what EOC or RCC teachings say, but Protestant teachings would say that not only can
we not fully understand it, but there's no reason to assume that Mary herself did either.
But Mary was content with that. She performed her assigned task for the kingdom with courage and faithfulness. She brought forth the Divine Word Incarnate into the World. She cared for him as any infant or toddler or child would need to be cared for. She nursed him, bathed him, fed him, protected him from dangers, ... and no doubt went down to the local Synagouge School from time to time to tell them they weren't challenging her boy enough.
Okay, maybe not that last part, but you get my meaning. (I can just see the teacher now. "Not challenging him?! How can we? It's like He was
there when the Torah was written!")
She was his mother, the mother of God Incarnate. What that means, I think, is unclear precisely because the nature of the Hypostatic Union is a mystery.
Even most Protestants believe she has a special place in the Kingdom. The Apostles brought forth the
message of the Gospel into the world. Mary brought forth the
embodiment of the Gospel into the world. In some respects, it is very appropriate to think of her as the
First Apostle.
So, is the title "Mother of God" appropriate? I don't see how it could be otherwise, at least on a certain level. What does it mean, exactly? What was the extent of her role in bringing forth Diety into the world? Well, since we can't understand the H.U., we can't really answer that question.
We should be content to understand what we can, and leave the rest to faith.
Romans 5:8 "But God demonstrates his own love for us in this: While we were still sinners, Christ died for us."