She is the mother of the incarnate Word, the Lord Jesus. This is just the point: Jesus is God. Jesus is a Person. Mary is the mother of a Person. Therefore Mary is the Mother of God.
"To refer to Mary as the mother of God implies that she precedes God."
That is nonsense. God existed from all eternity. Mary existed from, oh, I don't know, 20 BC or so. That is elementary. We all realize that.
"I doubt this error is lost on Rome, so we can only conclude Rome likes that error and wants to encourage it."
You really should desist from this kind of defamation. This is a violation of the Commandment which says "Thou shalt not bear false witness against they neighbor."
Christians have called Mary the Mother of God since Elizabeth did in Luke 1:43, a usage which was clarified in precise detail at the Council of Ephesus in 431 AD. No Catholic OR Orthodox pope, patriarch, priest, council, catechism, or any other Church teaching source has ever said that Mary "precedes" God or "existed before God" or any such thing.
The only distinction between Theotokos (God-birthgiver) and Theomētēr (Mother of God) is that the former emphasizes the birthmother (physical gestation) aspect of motherhood, while the later emphasizes the family relationship (one could say bloodline or genetic) aspect. Both are Biblical, both are true, since Mary was the mother of Christ our God in both the genetic and the gestational sense.
There is not, in either word, the slightest implication of "eternal" Motherhood, or that Mary is higher, or first, or existed before God.
If you can produce evidence to the contrary, I will send an apology to every single person on this thread.
Discussion is fine. Disagreement is fine. Dissent is fine. Factual misrepresentation --- slander --- is not fine. Desist from this.