That is a good observation on your part, and yes, Jesus is God based upon the substantial weight of Scriptural evidence, being ontologically the Son of God. But the issue is that of the careful use of language in ascribing titles in Scripture to God and man, with the objections to ascriptions to Mary including some which may be technically allowed, but are objectionable (as the link explained) as being unnecessary and inferring more than is written (1Cor. 4:6) as part of an exaltation of Mary which does just that.
Like as Ratzinger says “Co-redemptrix” departs “too great an extent from the language of Scripture and the Fathers” and gives rise to misunderstandings, so my objection here is that while “mother of God” may be technically allowed in the sense that she was the vessel through whom the body God had prepared (Heb. 10:5) came, and that she cared for the incarnated God to whom she owed her own existence, yet she is never actually called that, let alone that being a title, and it comes too close to inferring an ontological oneness with God.
As for Mary being called the “mother of my Lord,” whether Elisabeth was recognizing His Deity, or referring to Him in the Jewish Messianic sense as a ruler/deliverer from God, yet this is not needed for “mother of God” to be allowed in the aforementioned restricted sense. But again, the objection is to its warrant as part of a hyper exaltation of Mary not seen in Scripture.
In addition, even the distinguishing use of “Lord” and “God” attests to the careful use of titles, as without negating His Deity, the Holy Spirit typically makes a positional/ functional distinction between God and the Lord Jesus. For the “Lord” aspect mainly describes the Divine Christ's role as the administrator and judge of all, for while He is Divine, and one with the Father as being God, yet “God has made that same Jesus...both Lord and Christ,” reigning over all until He has put all enemies under his feet, and then shall the Son also himself be subject unto him that put all things under him, that God may be all in all.” (Acts 2:34-36; 1Cor. 15:25,28)
And as it is also technically true that Jesus was the son of David, that would make his wife and by extension Mary's mother, etc. one the grandmothers of God. But such formal titles are too prone to be misleading and are unnecessary, unless your intent is to exalt a person with as much attention and supererogatory of praise as possible, based upon extrapolations of some text (much in distinction to Paul, whose sufferings and labor of love the Holy Spirit abundant documents). And i am sure the humble and holy Mary of Scripture would refuse such inordinate devotion, versus the degree of attention and honor Scripture plainly gives her, and affirm, "I know not to give flattering titles; in so doing my maker would soon take me away. " (Job 32:22)
The problem with you understanding is that scripture itself disagrees with you. Thomas in addressing Jesus in the upper room: Joh 20:28 Thomas answered him, "My Lord and my God!"
Jesus was recognized as God just as John the Baptist recognized him in utero and why Elizabeth was able to address Mary as the Mother of God. all this was accomplished through the Holy Spirit.