What is the basis for Peter’s primacy? The early Acts of the Apostles, plus many confirmations from other popes and from medieval, post-Schism councils.
67. The Successor of Peter is thus, by the will of Christ, entrusted with the preeminent ministry of teaching the revealed truth. The New Testament often shows Peter “filled with the Holy Spirit” speaking in the name of all.”[Acts 4:8; cf. 2:14; 3:12] It is precisely for this reason that St. Leo the Great describes him as he who has merited the primacy of the apostolate.”[Cf. St. Leo the Great, Sermo 69, 3; Sermo 70, 1-3; Sermo 94, 3; Sermo 95 2: S.C. 200, pp. 50-52; 58-66; 258-260; 268] This is also why the voice of the Church shows the Pope “at the highest point- in apice, in specula- of the apostolate.”[Cf. First Ecumenical Council of Lyons, Constitution Ad apostolicae dignitaties: Conciliorum Oecumenicorum Decreta, ed. Istituto per le Scienze Religiose, Bologna 1973, p. 278; Ecumenical Council of Vienne, Constitution Ad providam Christi, ed. cit., p. 343; Fifth Lateran Ecumenical Council, Constitution In apostolici culminis, ed. cit., p. 608; Constitution Postquam ad universalis, ed. cit., p. 614; Constitution Divina disponente clementia, ed. cit., p. 638.] The Second Vatican Council wished to reaffirm this when it declared that “Christ’s mandate to preach the Gospel to every creature (cf. Mk. 16:15) primarily and immediately concerns the bishops with Peter and under Peter.”[Ad Gentes 38]
The full, supreme and universal power”[Lumen Gentium 22] which Christ gives to His Vicar for the pastoral government of His Church is this especially exercised by the Pope in the activity of preaching and causing to be preached the Good News of salvation.
Vatican II speaks more of the responsibility of the Bishop of Rome, rather than strictly what is owed to the Holy Father by others. In terms of evangelization, it is about what the pope does for others, how the pope serves the cause of evangelization, and for this topic, less the things that some Catholics find more exciting about the person or the role.