To say that “God-bearer” is more precise than “Mother of God” separates gestation/delivery from motherhood. There’s a lot of that going around these days, but rent-a-wombs weren’t a thing back in the day.
Evidently that is exactly the precision the early Church was looking for by using “Theotokos”: it affirms Mary’s role as God-bearer in her motherhood of Jesus in the flesh and in time, and avoids the misconception that she was somehow his mother from eternity, or the source of his Godhood.