The answer
Only the priest is supposed to give out the portions of the bread of blessing (the Eulogia) to the congregation when they leave the church after the end of the Mass and following the recitation of the final blessing.
When the believers receive this Eulogia from the hand which was only minutes before, touching he body of Christ, it has a better effect on their hearts and they can feel conscious in taking a blessing from the hand of the Father, from the hand of the priest of God.
Also, when the priest distributes the Eulogia, it gives him a chance to see who has attended the service, and who has not, so that he can ask after them and perhaps pay them a visit. Sometimes it provides him with an opportunity to say a few words to various people, and for them to speak to him. These moments could be used for any useful purpose, offering the chance for him to congratulate, to give his condolences, to encourage, to pray for someone, to arrange an appointment with someone to pray about something, or arrange a visit.
It is also an opportunity for the people to receive the blessing of the priest and to say hello to him before they leave the church.
The wholly consecrated deacon, however, is one of them, one of the clergy after all, but generally speaking, it is rare nowadays to find such deacons who are entitled to wear clerical dress, and who have devoted themselves completely to the ministry. Most of the deacons in the church are only less in rank :-aghnastus (reader) or Epideacon (assisting deacon).
But as far as distributing this Eulogia during the sharing of the holy sacraments is concerned, this is something definitely unsuitable, for it would mean that attention was being diverted from the divine mysteries to something else, when the only thing that should be going on at that time is a hymn of praise.
The expression you mentioned, to 'save time' is an unacceptable excuse because this is essentially a spiritual situation, which is important, and demands attention. Time, however, can be recouped in other ways. It isn't right for us to do wrong from the spiritual point of view on the flimsy pretext of 'saving time'!
This is like someone who leaves church during the service, and even during its holy moments, to 'save time'!!
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Thrice-Blessed Pope Shenouda III of Blessed Memory