The Leonine Prayers were meant to be recited after Low Mass for various specific intentions, originally the defence of the temporal sovereignty of the Holy See. Actually, even before Leo XIII, Pius IX had requested that three Hail Marys and a Salve Regina be recited after Low Mass, but only within the boundaries of the Papal States.
Pope Pius XI, following the Lateran Treaty of 1929, decided that the Leonine Prayers would be offered for a new intention: “the restoration to the afflicted people of Russia of the tranquility and freedom to profess the Catholic faith.” That was (and still is) frequently misinterpreted as a “Prayer for the Conversion of Russia”.
The final form of the Leonine Prayers consisted of three Hail Marys, a Salve Regina (followed by a versicle and response), a prayer for the conversion of sinners and the liberty and exaltation of Holy Mother Church, followed by the prayer to Saint Michael the Archangel. Pope Pius X allowed the addition of the invocation “Most Sacred Heart of Jesus, have mercy on us”, repeated three times.
The fulfillment of the declared purposes of these prayers–the securing of the temporal sovereignty of the Holy See and the freedom of Catholics in Russia to follow their consciences–has rendered the custom outdated, for one thing. More importantly, however, they were not meant to be recited after public Masses, only private Masses. Some say after “Low Masses”. The ordinary form of Mass, however, knows no level of celebration described as “Low Mass”.
At celebrations of the Tridentine Low Mass (the Extraordinary Form of Mass) the prayers are often said, but wrongly. They were officially suppressed by the Holy See in 1965.
There was a time not long ago that right after the dismissal, the church would erupt into a bunch of chattering and noise. Several of us led by an elderly lady would kneel and begin saying the Leonine prayers. Eventually folks became quieter and left church much more reverently. After the grand old lady passed on our little group has continued this practice.
It isn’t permitted to the laity to publicly lead the suppressed Leonine Prayers, in church, after Mass, when the pastor will not (because he may not). Additionally, there are those of us who remain after Mass to make prayers, meditations, or devotions who would be distracted and annoyed by the practice of a group of the laity imposing their prayer, out loud, upon everybody else.
For 45 years the Leonine Prayers have not been with us. They served their purpose at the time they were in vogue. They no longer have a purpose, today. There is no need to restore them.