In the Christian East, the practice of saying Our Father's and the Troparion to the Mother of God does indeed originate in the early centuries of the Church.
St Paul of Thebes reflected a well-established practice of walking along with two bags slung over his shoulders. In one bag there were 300 pebbles and he would place one in the empty bag on his other shoulder for each prayer he would recite.
In Ethiopia, monks from a VERY early time had squate notches on the top of their "prayer staffs" or "miquamia" for counting prayers. I have such a miquamia with 50 notches and you can stand with the staff in your hand with your thumb moving over the knotches as you pray. The Orthodox also have similar things in the form of wooden boards with knotches which resembles a scribbage board (is that the correct name for this game?).
It was St Pachomios of Egypt who tied "knotted cords" or like the prayer rope of today, every cord with 100 knots divided every 25, 33 or ten with a larger divider bead (our Master Beadsman, Phillip Rolfes, has developed amazing forms of these of which the Desert Fathers themselves would be justifiably proud - and who would undoubtedly want his email to order some from him directly . . .).
So we know the prayer cord, the knotted cord, was in vogue among monastics at the time of St Pachomios and also at the time of St Basil the Great who prescribed forms of the Jesus Prayer to be used on it for those monastics who could not read the psalms or the Divine Office. The prayer rope is, to this day, called in Orthodoxy the "Rosary of St Basil" (i.e. "Vervitsa").
So the Roman Catholic church is truly a Johnny come lately in this regard, if it relies only on unreliable stories in the lives of some of its later saints in this regard! ;)
Prayer to the Mother of God originated at the time of the Apostles, especially immediately after her Dormition and Assumption into heaven. She appeared in bodily form to the Apostles when they were at supper to confirm she had indeed been assumed bodily into heaven and told them to call on her with the prayer "Birthgiver of God, help us!" and she would hear and answer them, based on a promise given her by her Son Himself. This prayer has transformed into the oft-repeated invocation of the Byzantine East, "Most Holy Theotokos, save us!"
The Hail Mary in the East developed early and was a union of both scriptural and traditional elements: "O Theotokos Virgin, Rejoice Mary Full of Grace, the Lord is with Thee! Blessed are you among women and blessed is the Fruit of your Womb, for you have given birth to Christ the Saviour, the Redeemer of our souls!" (This is the UGCC and Old Believer version).
The prayer is therefore not primarily focused on the Virgin Mary, but on her as the Birthgiver of Christ, the Divine Son of God. Nestorius' "Christotokos" was opposed by the Orthodox Catholic "Theotokos" and the latter confession affirmed one was with the true Church of Christ.
Also the prayer "We fly to thy patronage" is of very old provenance.
This devotion was also reflected in the hymns of St John of Damascus, of words of St Herman of Constantinople (also quoted by Montfort) who said that God gives devotion to the Mother of God to those whom He wishes to save! etc.
The prayer cord of St Basil and St Pachomios was used to pray the Jesus Prayer, the Our Father, the Hail Mary (as it is still done on Mt Athos and elsewhere based on the most ancient books and traditions) and any other prayer we wish to repeat.
Monastics also memorized a single psalm, such as Psalm 23, and repeated it 150 times.