Yes, the Church has always allowed this, even before the Council. Deacons could distribute Holy Communion because they received the sacrament of Holy Orders. The Diaconal Ordination is sacramental; even though their hands are not consecrated, and they are not priests, they have still received the grace of ordination as deacons.
Thus, the Latin Church always considers them extraordinary ministers, not ordinary ones, because their hands are not consecrated. In the Oriental Churches, they usually do not touch the Blessed Sacrament or the holy host directly but instead hold the chalice and distribute Communion with it. In this case, they are not touching the Precious Blood directly. Therefore, it is the common tradition of the entire Church that only priests have the prerogative to directly touch the Body of Christ, except in exceptional cases, such as during persecution. In the past, the Church allowed laypeople to bring Holy Communion to the dying or to prisoners, but this was only permitted when the Holy Mass was celebrated outside, particularly during times of persecution.
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