Christmas as the birthday of the Church?

 From both a theological and a liturgical point of view, it is not unusual to associate the birth of the Church with Good Friday (particularly the flow of blood and water from the side of Christ) and with Pentecost (the gift of the Holy Spirit and the commencement of the public teaching of the Apostles).

But subtleties in the Roman liturgy of 31st December and 1st January prompt the association of the birth of the Church with Christmas. Firstly, the Office of Readings on 31st December, which contains a reading from a Sermon by St Leo the Great:

For the birth of Christ is the origin of the people of Christ, and the birthday of the head is the birthday of the body.

It is true that each of those who are called is allotted a particular place, and that all the children of the Church are separated from each other by intervals of time. However, just as all the faithful together, born of the waters of baptism, are crucified with Christ in this passion, raised with him in his resurrection, and given a place at the Father's right hand in his ascension, so too, with him they are born in this his birth.

The idea that the Virgin Mary is also the first to become a disciple of her Son, a thought suggested from Pope Paul VI's Apostolic Exhortation Marialis Cultus n.37, supports the association of Christmas with the birth of the Church:

 [She is] the perfect model of the disciple of the Lord: the disciple who builds up the earthly and temporal city while being a diligent pilgrim towards the heavenly and eternal city; the disciple who works for that justice which sets free the oppressed and for that charity which assists the needy; but above all, the disciple who is the active witness of that love which builds up Christ in people's hearts. 

And John Paul II wrote in his Encyclical Letter Redemptoris Mater n.26:

But above all, in the Church of that time and of every time Mary was and is the one who is "blessed because she believed"; she was the first to believe. From the moment of the Annunciation and conception, from the moment of his birth in the stable at Bethlehem, Mary followed Jesus step by step in her maternal pilgrimage of faith.

The second subtlety in the liturgy is the Prayer after Communion at Mass on 1st January, the Solemnity of Mary, the Holy Mother of God, which closely connects Mary's motherhood of her Son to her motherhood of the Church:

We have received this heavenly Sacrament with joy, O Lord:
grant, we pray,
that it may lead us to eternal life,
for we rejoice to proclaim the blessed every-Virgin Mary
Mother of your Son and Mother of the Church.