When is the birthday of the Church?

I have noticed that people tend to speak of the "birthday of the Church" at Pentecost? However, is this the case? In the Roman Missal the preface for the feast day uses the expression: "as the Church came to birth" rather than "was born." This speaks to me of the idea that the Church is being born in an ongoing manner, as new members are added through baptism. It can also take into account the pre-existence of the Church in the Second Person of the Holy Trinity and to be in line with Jesus Christ being the same yesterday, today and forever (Heb 13: 8). 

Furthermore, when we think of Our Lord, his birth is when, as it states in the creed, "he became man." This is not the case for the Church. In fact, with Pentecost it can be said that "she came to birth in the Spirit." As the Second Eve she finds her origin from the side of Christ and is joined to him in a common life "consortium totius vitae" as the Bride of Christ sharing in his ministry, Proverbs 31: 10-31 and Ephesians 5: 22-33, and is being continually renewed in time.

A suitable text for this idea might be: "The first man was from the earth, a man of dust; the second man is from heaven. As was the man of dust, so are those who are of the dust; and as is the man of heaven, so are those who are of heaven. Just as we have borne the image of the man of dust, we will also bear the image of the man of heaven." (1 Cor 15: 47-49)

This sense of becoming is also evident in the preface for the Solemnity of the Assumption of the Blessed Virgin Mary. We read there: "For today the Virgin Mother of God was assumed into heaven as the beginning and image of your Church's coming to perfection...". The Church is a "work in progress." She sees in Our Lady the fulfilment of what she will become when, as we read in Lumen Gentium, 2:

He (God) planned to assemble in the holy Church all those who would believe in Christ. Already from the beginning of the world the foreshadowing of the Church took place. It was prepared in a remarkable way throughout the history of the people of Israel and by means of the Old Covenant. In the present era of time the Church was constituted and, by the outpouring of the Spirit, was made manifest. at the end of time it will gloriously achieve completion, when, as we read in the Fathers, all the just, from Adam and "from Abel, the just one, to the last of the elect," will be gathered together with the father in the universal Church.

The danger of people contemplating the Church as being "born" or having a "birthday" is that they are going to fixate on the institution of the Church by the Lord in time as a solely historical event to which they are the heirs. The anamnesis of the Eucharist and Pentecost is a present reality which challenges the disciples to actualize the life giving action of the exitus and reditus of the Holy Trinity in their own time as you described in the perichoresis. 

Since the Eucharist is the memorial of the whole of Christ so too the life and nature of the Church embodies, albeit deficiently in her members, that reality. To this end the story and doctrine regarding Our Lady is the only true picture of what God wants for the Church as a whole. The feast of the Assumption can be seen as a Pentecostal event as lowly Miriam of Nazareth, through the power of her Son's resurrection, shares fully in his victory through the Holy Spirit. She realizes the destination of her humble faith by being crowned Queen of heaven and earth. Pentecost has approximately one hundred and twenty fearful and lowly individuals, transformed and infused by the Holy Spirit, to become a sign and sacrament of God's salvific power manifested in a community, the People of God, recreated perennially by Baptism and the Eucharist.