The Church takes seriously the prayers at the time of a Christian’s death. The first responsibility is that of any and all believers: not only to pray for the dying, but to be with them as a sign of unity and communion:
213. Christians have the responsibility of expressing their union in Christ by joining the dying person in prayer for God’s mercy and for confidence in Christ. In particular, the presence of a priest or deacon shows more clearly that the Christian dies in the communion of the Church. He should assist the dying person and those present with the recitation of the prayers of commendation and, following death, he should lead those present in the prayer after death. If the priest or deacon is unable to be present because of other serious pastoral obligations, other members of the community should be prepared to assist with these prayers and should have the texts readily available to them.
The rite continues, emphasizing the role of the ordained. A very serious responsibility is placed on priests and deacons. What sort of obligation would prevent a priest from attending to the dying? I can only think of two: a parish’s celebration of liturgy or another dying person.
I would suggest that a copy of the pastoral care rites should be ready for any pastoral minister–anyone who visits the sick, and certainly any lay ecclesial minister who has pastoral duties.
I know that the PCS series isn’t drawing much commentary, but I would consider an in-depth study of these rites virtually mandatory for seminarians, and any serious pastoral care minister.