It depends on whether you are talking about the linguistics of the title or the theology of the title. “Theotokos,” an early Greek name for Mary, is often translated to “Mother of God.” According to scholars, a more accurate translation would be “the one who gave birth to the one who is God.” So in that sense, “birthgiver” is probably a more linguistically accurate title, at least if you look at the original Greek.
On a theological level, though, “Mother of God” is perfectly correct. If Jesus was both fully God and fully human, and Mary was his mother, then yes, it is accurate to call Mary the Mother of God. This does not mean that she existed before God; rather, it means that she was the mother of Jesus Christ, the second person of the Trinity.
And when you compare the broader meaning of the two titles, “Mother of God” more fully reflects the active role that Mary played in the life of Jesus. She was not just a passive channel through which Jesus came into the world; instead, she fed him, bathed him, taught him, nurtured his faith, stood by him during the worst times, and did all the things most moms do for their kids. In this way, “Mother of God” reminds us that Mary’s role did not stop with the Nativity, but has continued ever since.