I believe if we read Matthew 16:18, John 21:6, Luke 22:31 and Matthew 7:24 together we will get a more clear picture of what our Lord Jesus wanted.
In the absence of His human incarnation Jesus wanted someone (a human) to lead His physical Church and tend His whole flock In John 21:16 It becomes very clear: "16 He says to him again: Simon, son of John, do you love me? He said to him: yea, Lord, you know that I love you. He said to him: Feed my lambs. 17 He said to him the third time: Simon, son of John, do you love me? Peter was grieved because he had said to him the third time: Do you love me? And he said to him: Lord, you know all things: you know that I love you. He said to him: Feed my sheep"
Btw, the Greek words that are translated as "feed" also mean tend or "take care of" ποίμαινε τὰ προβάτιά μου; βόσκε τὰ προβάτιά μου
Here, our Lord charges Peter with the superintendency of all his sheep, without exception; and consequently of his whole flock, that is, of his own Church. We can argue who the successor of Peter is today, but Jesus in John 21 does leave a single successor, not a big group of bishops working separately from their ethnic silos.
Matthew 16: 18 And I say to you: That you are Peter; and upon this rock I will build my church, and the gates of hell shall not prevail against it. In Greek "κἀγὼ δέ σοι λέγω ὅτι σὺ εἶ Πέτρος, καὶ ἐπὶ ταύτῃ τῇ πέτρᾳ οἰκοδομήσω μου τὴν ἐκκλησίαν, καὶ πύλαι ᾅδου οὐ κατισχύσουσιν αὐτῆς"
Upon this rock, etc... The words of Christ to Peter, spoken in the vulgar language of the Jews which our Lord made use of, were the same as if he had said in English, Thou art a Rock, and upon this rock I will build my church. So that, by the plain course of the words, Peter is here declared to be the rock, upon which the church was to be built: Christ himself being both the principal foundation and founder of the same. Where also note, that Christ, by building his house, that is, his church, upon a rock, has thereby secured it against all storms and floods, like the wise builder, (see Matthew 7:24-25.):
24 Every one therefore that hears these my words, and does them, shall be likened to a wise man that built his house upon a rock 24 Πᾶς οὖν ὅστις ἀκούει μου τοὺς λόγους [τούτους] καὶ ποιεῖ αὐτοὺς ὁμοιωθήσεται ἀνδρὶ φρονίμῳ, ὅστις ᾠκοδόμησεν αὐτοῦ τὴν οἰκίαν ἐπὶ τὴν πέτραν. In Matthew 7:24 and in Matthew 16:18 the Greeks words build (οἰκοδομήσω) and rock (Πέτρος, πέτρᾳ) are repeated.
So Jesus selected Peter to be the foundation of His Church due to Peter's resemblance to the wise man who heard Jesus' words and did (ποιεῖ ) them
Finally in Luke 22:31-33, Satan singles out Peter for an attack and Jesus gives special protection, so Peter can support the flock: "31 And the Lord said: Simon, Simon, behold Satan has desired to have you, that he may sift you as wheat. 32 But I have prayed for you, that your faith fail not: and you, being once converted, support your brethren" Peter is the only Apostle that Jesus prays for for special protection from Satan, so he can lead His flock