Some of what follows is sheer speculation, thinking out loud. If it conflicts in any way with established doctrine and understanding, it should be disregarded, and I would greatly appreciate a note correcting any such error.
Mary, Queen and mother of Carmel and big sister to the Carmelites and to all contemplatives. From earliest times, Carmelites have viewed Mary as both Queen and Mother and as true Sister and exemplar of the Christian expression of St. Elijah. In a certain way, she is the Mother Superior of the Order, chief among the sisters and brothers--example and guide for the attentive.
Also from earliest times, Carmelites have had a special devotion to Mary. The earliest manifestation of this was in the primitive Oaths and Vows that referred to the Carmelite follower of Mary as Vassal and Fief of Mary--the true property and servant, the one owed protection and special care of the Blessed Virgin. Even today, the Carmelite, with his or her habit of the brown scapular, claims the special attention of Mary. (Which is, in no way to imply favoritism on the part of the Blessed Virgin, it is merely reflective of the origin of the Order and its charism.)
True devotion to Mary does not consist of endless prayers to her but of substantive imitation of her way of life and of obedience to her very few direct words to us.
1 On the third day there was a marriage at Cana in Galilee, and the mother of Jesus was there;
2 Jesus also was invited to the marriage, with his disciples.3 When the wine failed, the mother of Jesus said to him, "They have no wine."
4 And Jesus said to her, "O woman, what have you to do with me? My hour has not yet come."
5 His mother said to the servants, "Do whatever he tells you."
Do whatever he tells you. These are the words of the Mother and sister who already has reason to know that what is being done is extraordinary. As she pondered the events of her life in an extended thirty year examen, she came to know who and what Jesus is even before there has been any overt sign. It is at a word from her that the prophetic and salvific mission begins. It is as though the Holy Spirit in both unites them at this unique time and place to initiate the Earthly preaching mission of Jesus. At Mary's word, the every obedient, loyal, and loving Son is released just as He had been bound after the finding in the temple.
One of the chief ways in which devotion to the Blessed Virgin is expressed is through praying the Rosary. In the before times, long ago, the Rosary was a device that led to a kind of extended lectio without the necessity of being able to read. One pondered the mysteries of the life of the Blessed Virgin and of Jesus Christ in the course of praying through the Rosary. In addition, the Rosary was a kind of "replacement" for the Liturgy of the Hours for those who could not read. It became possible through the three sets of mysteries of the Rosary to pray through the 150 psalms of the psalter.
Of the rosary, Pope John Paul the Great, of recent memory, wrote:
from the Apostolic Letter "Rosarium Virginis Mariae"
 With the Rosary, the Christian people sits at the school of Mary and is led to contemplate the beauty on the face of Christ and to experience the depths of his love. Through the Rosary the faithful receive abundant grace, as though from the very hands of the Mother of the Redeemer. . .
 I have felt drawn to offer a reflection on the Rosary, as a kind of Marian complement to that Letter and an exhortation to contemplate the face of Christ in union with, and at the school of, his Most Holy Mother. To recite the Rosary is nothing other than to contemplate with Mary the face of Christ.
[T]he most important reason for strongly encouraging the practice of the Rosary is that it represents a most effective means of fostering among the faithful that commitment to the contemplation of the Christian mystery which I have proposed in the Apostolic Letter Novo Millennio Ineunte as a genuine “training in holiness”
 The contemplation of Christ has an incomparable model in Mary. In a unique way the face of the Son belongs to Mary. It was in her womb that Christ was formed, receiving from her a human resemblance which points to an even greater spiritual closeness. No one has ever devoted himself to the contemplation of the face of Christ as faithfully as Mary. The eyes of her heart already turned to him at the Annunciation, when she conceived him by the power of the Holy Spirit.
 The Rosary is both meditation and supplication. Insistent prayer to the Mother of God is based on confidence that her maternal intercession can obtain all things from the heart of her Son. She is “all-powerful by grace”, to use the bold expression, which needs to be properly understood, of Blessed Bartolo Longo in his Supplication to Our Lady.This is a conviction which, beginning with the Gospel, has grown ever more firm in the experience of the Christian people. The supreme poet Dante expresses it marvellously in the lines sung by Saint Bernard: “Lady, thou art so great and so powerful, that whoever desires grace yet does not turn to thee, would have his desire fly without wings”. When in the Rosary we plead with Mary, the sanctuary of the Holy Spirit (cf. Lk 1:35), she intercedes for us before the Father who filled her with grace and before the Son born of her womb, praying with us and for us.
I won't belabor the point. The entire letter is worthy of careful consideration--it may be among the most Carmelite of the Letters of this most famous Third Order Carmelite. The understanding of both the Rosary and of what it teaches, strikes me as profoundly Carmelite. We don't recite the prayers of the Rosary as a rote exercise or as a devotion, we pray the Rosary as a model and a source, a root, as it were, of contemplation. For the Carmelite, any other use of the Rosary falls short of its true potential AND, more importantly, falls short of true devotion to Mary. True devotion to Mary, in the Carmelite tradition, consists in imitating her to the extent possible according to our way of life and our present cultural milieu. Yes, through intercession and prayer, we trust her with all of our concerns, but that falls short of the perfection of devotion, which consists of Imitating her, and in the imitation of Her, gazing on and becoming like Her Son. In a very real way, in her thirty years of meditation upon the mystery of her life and the Incarnation, she bound herself to her Son--as the Spouse of the Holy Spirit, she already experienced the "spiritual marriage" and "mystical union." In some way that I don't comprehend or presume to explain, it would seem to me that she shared in the sufferings of Christ in His passion AND carried her own weight of suffering (as a Mother losing a beloved child) as well. In the depths of the mystery of the Passion, she seems to play two roles--one in union with the Holy Trinity through the indwelling Holy Spirit and the complete consummate spiritual union, the other as sorrowing mother, observer and witness of the trials, terrors, and horrors, of the Passion. (I hope I don't overstate the case here, forgive me if I have or if I have inadvertently written any error in regard to these deep mysteries. They are truly beyond me, and I hope I do not go beyond what the Holy Catholic Church teaches. Here most of all, I humbly await and accept correction.)
Thus, the Carmelite looking upon the Blessed Virgin sees both contemplative and example. She is Queen and Mother of Carmel. She is the chief protector, guide, and example of the Order. But by virtue of her human birth She is our sister as well as our mother in faith. This is not so odd as it sounds--in many religious order, the Mother Superior, is merely the chief of all the sisters. After her term of office, she returns to the state she had before in the Order. Mary is simply the permanent Mother Superior of all Carmelites.
I hope I've provided some insight into the role and importance of the Blessed Virgin in Carmelite devotion. It explains why a great many Carmelites had difficulty with reciting the Rosary on a regular basis. The common recitation of it does not often lend itself to the depth implied by John Paul the Great in his letter. Too often it is too easy to be carried along on the tide of the familiar and not enter into the depths of what is available in this most wonderful of devotions. Truly prayed, the Rosary should effect a profound change in the pray-er making her or him more like the subject of the devotion and more like Jesus Christ. Too often, the Rosary is a chain of supplication and intercession more than it is an entrance into the depth of the life of Our Savior and His Mother. But, as Saint Teresa of Avila points out, even vocal prayer is raised to the level of mental prayer if we keep in mind always the vastness of great dignity of the One to Whom we speak. And even though we seem to speak to the Blessed Virgin, the Rosary is a continual plea to God through the merciful intercession of the Blessed Virgin. A properly prayed Rosary, faithfully accomplished every day, is as much a gateway to contemplation as faithful following of the Liturgy of the Hours or Lectio Divina. That the latter two along with special devotion to the Blessed Virgin--either in the form of the Rosary or in other special devotions--make up the pillars of the introduction to prayer in Carmel should come as no surprise. That they serve as the gateway to meditation, contemplation, and as God wills, eventual union with God, again should not be the source of any surprise. The Blessed Virgin Mary looks with an eye of special kindness on those who wear her scapular worthily and upon those who invoke her aid in learning to look upon the face of Her son. This is true whether one is Carmelite or not. Carmelite Spirituality merely shows these forth for what they are in a way unique to the Carmelite Order. They are a special gift to the Carmelites and hence to the Church at large--available for anyone who chooses to follow them within the order or outside. The Blessed Mother will not withhold the graces she bestows for the sake of a name.