But Mary was only the "source" of the 2nd person of the Trinity insofar as He was Man. So why not call her that? Why insist on calling her Mother of God? She didn't give God His beginning. She was mother of Jesus as Man (even while His deity was left intact). The Bible calls her Mother of Jesus. It does not call her Mother of God. This does not in any way take away from Jesus' deity. It does however keep Mary from being put in the potentially misleading position of preexisting Christ as God.
Because to state it as you did, leads to what the council was called for in the first place. It was to confront two types of heresies. The first that Jesus Christ was in reality to individuals, a man and God (this later was called Nestorian). The second was that he was either man or God, but not both. To call Mary only the mother of the human Jesus, but not the second person in the Trinity, implies that Jesus was not true man and true God in one individual. So, for the Incarnation to be what most orthodox (lower case o) Christians believe it to be, Mary had to be the mother of God, but she is not any type of deity herself. To state that she was only the mother of the man Jesus splits the person of Christ.
I will admit, the title "Mother of God" can lead to some rather disastrous excesses in folk religion, but to go the other way leads to saying that Jesus was not really true Man and true God. Kosta probably has a link to the council that decreed the title "Mary Mother of God", and I would suggest reading a book on the early Church theology like this one by Jarslov Pelikan.
http://www.amazon.com/Christian-Tradition-Development-Doctrine-Emergence/dp/0226653714/ref=pd_bxgy_b_text_b/104-6132202-4845522
It walks you through the early Christlogical controversies and why some of the things were phrased they way they are.