Many Catholics tend to use Mary being the Mother of God as a justification for all the other Marian dogmas and doctrines of Catholicism, especially the Immaculate Conception (e.g., Christopher Ferrara used Ephesus [431 AD] proclaiming Mary being Theotokos as evidence for the Immaculate Conception in his debate against James White).
While Mary can be understood as the mother of God as she was the mother of Jesus, a divine person, and even the 1830 Book of Mormon referred to Mary as the “mother of God” (e.g., 1 Nephi 11:18), there is nothing in Mary being Mother of God or Theotokos (literally “God-bearer”) and/or the decree of the Council of Ephesus that necessitate the other dogmas and doctrines.
Max Thurian, then a Reformed Protestant with strong ecumenical leanings who would later convert to Roman Catholicism, was correct when he wrote:
The Ephesian dogma has an essentially Christological import. Mary is not called Mother of God in order that her person may be glorified, but for the sake of Christ, in order that the truth about Christ’s person should be plainly and clearly understood. (Max Thurian, Mary Mother of all Christians [trans. Neville B. Cryer; New York: Herder and Herder, 1963], 74)
For more on Mariology, see my book:
The other major dogmas and doctrines of Mariology that divide Latter-day Saints and Roman Catholics are discussed, including two chapters on the Immaculate Conception.