It's a poorly named doctrine. Co-Redemtrix implies equality, that she played an equal role with Christ. If it was made dogma with that name, the average Catholic would think such, and that understanding is heretical.
As mentioned, (1) Co-Redemptrix of the world/Reparatrix of the whole human race has already been used by many Popes. (2) Co-Redemptrix does not imply equality at all, but sub-ordinate co-operation. St. Alphonsus calls the Blessed Mother of God the "Co-Operatress in our Redemption, because by Her Sorrows and Dolours at the Foot of the Cross, She Co-operated in giving us new birth.
Theologians explain the sub-ordinate co-operation of Mary in Redemption by the distinction between condign and congruous merit and satisfaction. Condign merit is merit in strict justice, such as only Jesus Christ merited. Congruous merit is merit based on friendship or charity, and in this way the Blessed Mother merited for us. Thus, Fr. Garrigou Lagrange, citing Pope St. Pius X, explains in the OP, "For the same reason, all that Christ merited for us on the Cross in strict justice, Mary merited for us by congruous merit, based on the charity that united her to God. Christ alone, as head of the human race, could strictly merit to transmit Divine life to us. But Pius X sanctioned the teaching of theologians when he wrote: "Mary, united to Christ in the work of salvation, merited de congruo for us what Christ merited for us de condigno."
Pope Leo XIII is just one of many Popes who have taught Co-Redemptrix. Here's an Encyclcal on it, specially suited for this month of October devoted to the Rosary, From: http://www.newadvent.org/library/docs_le13is.htm
Iucunda Semper Expectatione
On the Rosary
His Holiness Pope Leo XIII
September 8, 1894
To the Partiarchs, Primates, Archbishops, Bishops, and other Ordinaries in Peace and Communion with the Apostolic See.
Venerable Brethren, Greeting and Apostolic Benediction.
IT IS ALWAYS with joyful expectation and inspired hope that We look forward to the return of the month of October. At Our exhortation and by Our express order this month has been consecrated to the Blessed Virgin, during which for some years now the devotion of her Rosary has been practiced by Catholic nations throughout the world with sedulous earnestness. Our reasons for making this exhortation We have made known more than once. For as the disastrous condition of the Church and of Society proved to Us the extreme necessity for signal aid from God, it was manifest to Us that that aid should be sought through the intercession of His Mother, and by the express means of the Rosary, which Christians have ever found to be of marvelous avail. This indeed has been well proved since the very institution of the devotion, both in the vindication of Holy Faith against the furious attacks of heresy, and in restoring to honor the virtues, which by reason of the Age's corruption, required to be rekindled and sustained. And this same proof was continued in all succeeding ages, by a never failing series of private and public benefits, whereof the illustrious remembrance is everywhere perpetuated and immortalized by monuments and existing institutions. Likewise in Our age, afflicted with that tempest of various evils, it is a joy to Our soul to relate the beneficent influence of the Rosary. Notwithstanding all this, you yourselves, Venerable Brethren, behold with your own eyes the persistence--nay, the increase--of the reasons for renewing again this year Our summons to the Faithful to turn with increased ardor in prayer to Mary, the Queen of Heaven. Besides, the more We fix Our thoughts upon the character of the Rosary, the clearer its excellence and power appear to Us. Hence, while Our wish increases that it may flourish, Our hope grows also that through Our recommendation it may come to be more greatly prized, its holy use become more extended and flourish abundantly. But We shall not now return to the various instructions which in past years We have given upon this subject. We shall take instead the opportunity of pointing out the particular ruling and designs of Providence which ordains that the Rosary should have new power to instill confidence into the hearts of those who pray, and new influence to move the compassionate heart of Our Mother to comfort and succor Us with the utmost bounty.
2. The recourse we have to Mary in prayer follows upon the office she continuously fills by the side of the throne of God as Mediatrix of Divine grace; being by worthiness and by merit most acceptable to Him, and, therefore, surpassing in power all the angels and saints in Heaven. Now, this merciful office of hers, perhaps, appears in no other form of prayer so manifestly as it does in the Rosary. For in the Rosary all the part that Mary took as our co-Redemptress comes to us, as it were, set forth, and in such wise as though the facts were even then taking place; and this with much profit to our piety, whether in the contemplation of the succeeding sacred mysteries, or in the prayers which we speak and repeat with the lips. First come the Joyful Mysteries. The Eternal Son of God stoops to mankind, putting on its nature; but with the assent of Mary, who conceives Him by the Holy Ghost. Then St. John the Baptist, by a singular privilege, is sanctified in his mother's womb and favored with special graces that he might prepare the way of the Lord; and this comes to pass by the greeting of Mary who had been inspired to visit her cousin. At last the expected of nations comes to light, Christ the Savior. The Virgin bears Him. And when the Shepherds and the wise men, first-fruits of the Christian faith, come with longing to His cradle, they find there the young Child, with Mary, His Mother. Then, that He might before men offer Himself as a victim to His Heavenly Father, He desires to be taken to the Temple; and by the hands of Mary He is there presented to the Lord. It is Mary who, in the mysterious losing of her Son, seeks Him sorrowing, and finds Him again with joy. And the same truth is told again in the sorrowful mysteries."
Claudel, thanks, that's a reasonable objection. I hope the foregoing addresses it. We know some people misunderstand "Immaculate Conception" to mean Immaculate Conception of Jesus by Mary, rather than the Immaculate Conception of Mary Herself. We will have to explain it to those who misunderstand.
Faithful Catholics all believe this anywayPerhaps all faithful Catholics would be ready to believe it, but many right now have not even heard that Mary is Mediatrix of all Graces. Saints like Catherine of Sienna, St. Bernard, St. Alphonsus, St. Padre Pio, St. Maximillian Kolbe etc lay great emphasis on believing it, living it, teaching it. Mary dispenses all graces from first justification to final perseverance. It is certainly definable dogma, implied in Luk 1:28 itself, the Angelic Salutation, as well as in passages like Sir 24:25 which the Church applies to Her.
making things harder for ProtestantsThe intention is not to make things hard for Protestants, but for Catholic Christians to grow in love, devotion and gratitude to our Mother, for Her sufferings, dolours and sorrows all Her life and especially at the foot of the Cross in Union with Christ Crucified. St. Alphonsus: "But if Mary's lips were silent, her heart was not so, for she incessantly offered the life of her Son to the Divine Justice for our salvation. Therefore we know that by the merits of her dolours she cooperated in our birth to the life of grace; and hence we are the children of her sorrows. "Christ," says Lanspergius, "was pleased that she, the cooperatress in our redemption, and whom He had determined to give us for our Mother, should be there present; for it was at the foot of the cross that she was to bring us, her children, forth." If any consolation entered that sea of bitterness, the heart of Mary, the only one was this, that she knew that by her sorrows she was leading us to eternal salvation, as Jesus Himself revealed to Saint Bridget: "My Mother Mary, on account of her compassion and love, was made the Mother of all in heaven and on earth." And indeed these were the last words with which Jesus bid her farewell before His death: this was His last recommendation, leaving us to her for her children in the person of Saint John: "Woman, behold thy son." From that time Mary began to perform this good office of a Mother for us" Source: From the Glories of Mary, On the Dolours of Mary, by St. Alphonsus: On the Fifth Dolor