In the 1950s, sex outside of marriage was a social taboo. A single woman who became pregnant would have been abandoned by her family and ignored by the state. This was true in Catholic Ireland, Protestant England and Secular France. Had it not been for agencies associated with the Catholic and Anglican Churches, unmarried mothers would have been left to beg on the streets. This might seem strange 60 years later when the ideas of sex, marriage and conception have been separated, but don't try to hang modern morality (or lack of it) on 1950s society.
Yes, the story was only possible because of the Catholic Church. Without them, Philomena and her son would have not survived to tell their story.