This is no longer the position of the Church. Canon Law and Bl. John Paul II are very explicit. The distribution of the Eucharist belongs to the ordained. The deacon, once incardinated into a diocese or a religious order, which usually happens at ordination, can distribute communion without permission from a priest or bishop. It is already granted by Canon Law and the permission to do so in a particular jurisdiction is granted by the ordinary.
The only restriction, if it can be called that, is that the priest must also distribute Holy Communion. He cannot sit and let the deacon take over. If the congregation does not warrant two ministers of Holy Communion, the priest who presides must be the one to distribute. In reality, this law is meant to guide the priest, not the deacon.
The official position of the Church on this question is no longer that of the Council of Trent as it was written then. It was modified to “the ordinary minister of Holy Communion is the deacon, priest and bishop,” by Bl. John Paul II in 1983. It does not limit the distribution to the priest, except as stated above, which is in another canon. Popes always trump councils on all matters except dogmas and morals. Even when quoting the Council of Trent, we must always ask, “What is the current position of the Church?” The answer will usually be found in Canon Law, the CCC, or one of the papal writings.
1570 Deacons share in Christ’s mission and grace in a special way. The sacrament of Holy Orders marks them with an imprint (“character”) which cannot be removed and which configures them to Christ, who made himself the “deacon” or servant of all. Among other tasks, it is the task of deacons to assist the bishop and priests in the celebration of the divine mysteries, above all the Eucharist, in the distribution of Holy Communion, in assisting at and blessing marriages, in the proclamation of the Gospel and preaching, in presiding over funerals, and in dedicating themselves to the various ministries of charity.
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Can. 910 §1. The ordinary minister of holy communion is a bishop, presbyter, or deacon.
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As to your question about why the Franciscan history. It has nothing to do with competition. Actually, the two orders agree on this point. I was trying to show how this question is not new and why Aquinas position was never adopted as de fide.
Fraternally,
Br. JR, FFV