The term "mother of God" implies way more than the catholic wants to limit it to.
As you correctly note, the Father is God and the Holy Spirit is God.
If Mary is the "mother of God", then she is the mother of the Father as well and the Holy Spirit.
Catholics would do well to use the terms the Holy Spirit inspired the writers of the NT to use.
Mary, His mother (Matt 2:11,13,14,20,21)
...by the cross of Jesus were His mother...Jn 19:25)
Mary the mother of Jesus (Acts 1:14)
Here’s a yes-or-no question:
Is Jesus God the Son, the Second Person of the Trinity?
Is Jesus also God the Father and God the Holy Spirit? No. In the context Thomas' declaration of "My Lord and my God!" is only referring to him as God the Son. Likewise, the title of "Mother of God" is only referring to her motherhood of Jesus, God the Son, who is the Son of God and the Son of Mary. Not really hard to understand if you want to.