Why is today Pope St. Gregory the Great's feast in the Novus Ordo calendar and not Pope St. Pius X's, as it is in the traditional calendar?
Yes, they moved Gregory the Great's feast day from the day of his death, 3/12, to the beginning of his pontificate, 9/3.
Was that ever done before? I thought saints' feast days were always celebrated on the anniversary of his/her death (ie. the beginning of their life eternal).
While it is traditional to commemorate a saint on the anniversary of their death, that may not always be possible because of conflicts on the calendar.
St. Pius V died on May 1, 1572. He was commemorated on May 5 on the 1960 and previous calendars on May 5 as from the 6th century through 1954 May 1 had been the feast of SS. Philip and James, Apostles. In 1955 Pope Pius XII moved the feast of SS. Philip and James to May 11 and created the observance of St. Joseph the Worker on May 1.
The 1969 calendar moved the feast of these apostles to May 3 (back closer to their origional feast day of May 1) and the commemoration of St. Pius V to April 30, a date closer to his actual death.
St. Pius X died on August 20, 1914 and was canonized in 1954. On his revised Roman Calendar of 1955 Pope Pius XII commemorated Pius X on September 3, though I no of no significance of that date to his life.
St. Pius X's commemoration was moved to August 21 on the 1969 calendar, one day before the date of his death, as August 20th is the feast day of Saint Bernard of Clairvaux, Abbot and Doctor of the Church, who also died on August 20 and was cannonized in 1174.
The reason I have seen given for moving St. Gregory the Great's commemoration to September 3 is to not have it always fall during Lent, allowing for a more solemn liturgical commemoration, according to the current rubrics. Why St. Pius X was origionally assigned to September 3 rather than a date closest to his death on August 20, I have no idea.