Whatever the actual justification (whether one agrees with it or not) for using the phrase "Mother of God", I suspect a lot of objection to the term can be traced to the idea that "motherhood" precedes fatherhood, or even precedes, in divine terms, the begotten. (And I would say this idea can often be held as a kind of subconscious a priori assumption so that the person holding it may not even be aware that it is being assumed but simply draws conclusions without realizing it is being used in the process).
So, with this assumption, there is an inherent conclusion that if God has a mother then the mother must somehow be "above" God and therefore has more claim to be "God" herself than the supposed god she is claimed to be the mother of.
I don't know if that assumption is part of your internal process (conscious or not) and objection to the phrase, but I would just make clear that this is not what the Catholic view is. Even though some religions might use "motherhood" as the root (eg "Mother Earth" and such), Christianity (and Catholicism in particular)sees it as a created thing. Genesis even shows the Eve as being brought out of the Adam and not the other way around as we might narrowly and naively "scientifically" imagine if we were devising a creation story ourselves. (As it is, even in evolutionary thought, "motherhood" came rather late into the game I believe.)
So however God can, in his mysterious way, cause his very creation to "beget" his nature (talk about circular references and tail-eating serpents!
), the phrase "Mother of God" is, for me (as an admittedly biased Catholic who is obliged to accept it anyway), a perfectly reasonable one, both in terms of accepting the doctrine on Church authority, but also simply in terms of it not seeming self-contradictory or meaningless. One may have other objections, but if one can wriggle around the assumption of motherhood preceding fatherhood and the divine as I have suggested above, it is at least something one can "work with".
By the way, I would also note that even though you mention DNA in the first paragraph, in the second paragraph you write "Clearly Mary provided the nurturing womb in which the infant Jesus grew..." and perhaps you meant for the DNA reference to carry over. And if so, that is good. But because it is separated from that reference, and because it seems to be another of those "unconsciously held" assumptions held by many people that Mary was simply a sort of "conduit" or "tube" which Jesus "slid through" to get from his heavenly "place" down to earth, I would clarify that Christ's humanity was "woven" (if I can use that term) from the very DNA of Mary so that she was an integral part of his being and not just a "carrier" of the divine fetus as though she were some kind of surrogate mother. You may not have intended that meaning, but it is easy for others to miss that so I wanted to clarify at least what the orthodox intention is.
--Stanley
…on a night of rain Frodo smelled a sweet fragrance on the air and heard the sound of singing that came over the water. And then it seemed to him that as in his dream in the house of Bombadil, the grey rain-curtain turned all to silver glass and was rolled back, and he beheld white shores and beyond them a fair green country under a swift sunrise.