For a parish with multiple priests, or at a concelebrated Mass, it may make sense for one priest to take time away from pastoral contact with the people. Otherwise, GIRM 163 comes off as a bit too fussy:
163. When the distribution of Communion is over, the Priest himself immediately and completely consumes at the altar any consecrated wine that happens to remain; as for any consecrated hosts that are left, he either consumes them at the altar or carries them to the place designated for the reservation of the Eucharist.
Upon returning to the altar, the Priest collects the fragments, should any remain, and he stands at the altar or at the credence table and purifies the paten or ciborium over the chalice, and after this purifies the chalice, saying quietly the formula Quod ore sumpsimus, Domine (What has passed our lips), and dries the chalice with a purificator. If the vessels are purified at the altar, they are carried to the credence table by a minister. Nevertheless, it is also permitted to leave vessels needing to be purified, especially if there are several, on a corporal, suitably covered, either on the altar or on the credence table, and to purify them immediately after Mass, after the Dismissal of the people.
164. After this, the Priest may return to the chair. A sacred silence may now be observed for some time, or a Psalm or other canticle of praise or a hymn may be sung (cf. no. 88).
Either silence or a hymn of praise? What about both?
165. Then, standing at the chair or at the altar, and facing the people with hands joined, the Priest says, Let us pray; then, with hands extended, he recites the Prayer after Communion. A brief period of silence may precede the prayer, unless this has been already observed immediately after Communion. At the end of the prayer the people acclaim, Amen.
This post-Communion prayer is not a “closing prayer,” and when offered at this moment, before announcements and such, is a proper conclusion to the Liturgy of the Eucharist.
Neither the Order of Mass nor the GIRM prescribe the people’s posture during this prayer; only the priest is indicated to stand. My suggestion is that this prayer is prayed while people are seated. Brief announcements follow after the prayer, not before it.