That is what your church wants you to believe... but it is not retelling the entire Scripture passage! In the same passage, Peter is rebuked by Christ! (23Jesus turned and said to Peter, "Get behind me, Satan! You are a stumbling block to me; you do not have in mind the things of God, but the things of men.")
Matthew 20:1ff
1"For the kingdom of heaven is like a landowner who went out early in the morning to hire men to work in his vineyard. 2He agreed to pay them a denarius for the day and sent them into his vineyard.
3"About the third hour he went out and saw others standing in the marketplace doing nothing. 4He told them, 'You also go and work in my vineyard, and I will pay you whatever is right.' 5So they went.
"He went out again about the sixth hour and the ninth hour and did the same thing. 6About the eleventh hour he went out and found still others standing around. He asked them, 'Why have you been standing here all day long doing nothing?'
7" 'Because no one has hired us,' they answered.
"He said to them, 'You also go and work in my vineyard.'
8"When evening came, the owner of the vineyard said to his foreman, 'Call the workers and pay them their wages, beginning with the last ones hired and going on to the first.'
9"The workers who were hired about the eleventh hour came and each received a denarius. 10So when those came who were hired first, they expected to receive more. But each one of them also received a denarius. 11When they received it, they began to grumble against the landowner. 12'These men who were hired last worked only one hour,' they said, 'and you have made them equal to us who have borne the burden of the work and the heat of the day.'
13"But he answered one of them, 'Friend, I am not being unfair to you. Didn't you agree to work for a denarius? 14Take your pay and go. I want to give the man who was hired last the same as I gave you. 15Don't I have the right to do what I want with my own money? Or are you envious because I am generous?'
16"So the last will be first, and the first will be last."
Your point is a disaster and very WRONG.
First, you missed the entire reason why Christ ever came to us as Savior and you forgot to mention the particular context here as it relates to Peter and us all. Many here would be driven off a cliff to humanism if they took your conclusions of the gospel as Gospel IMO. :-)
Peter's failings and Christ's rebuke of him were BEFORE Christ sacrificed Himself for the sins of the world on the Cross and BEFORE the Holy Spirit came upon the Apostles to give them the full Armor of God.
Christ's established His Church and the Authority of the Church here on earth.
Christ is not made false or a liar because Peter was sinning had not yet received the benefit of Christ's sacrafice on the Cross or the Holy Spirit.
Here is a better account of Biblical history for you.
Peter becomes Head of the Apostles. In especially solemn fashion Christ accentuated Peter's precedence among the Apostles, when, after Peter had recognized Him as the Messias, He promised that he would be head of His flock. Jesus was then dwelling with His Apostles in the vicinity of Caesarea Philippi, engaged on His work of salvation.
As Christ's coming agreed so little in power and glory with the expectations of the Messias, many different views concerning Him were current. While journeying along with His Apostles, Jesus asks them: "Whom do men say that the Son of man is?" The Apostles answered: "Some John the Baptist, and other some Elias, and others Jeremias, or one of the prophets". Jesus said to them: "But whom do you say that I am?" Simon said: "Thou art Christ, the Son of the living God".
And Jesus answering said to him: "Blessed art thou, Simon Bar-Jona: because flesh and blood hath not revealed it to thee, but my Father who is in heaven. And I say to thee: That thou art Peter [Kipha, a rock], and upon this rock [Kipha] I will build my church [ekklesian], and the gates of hell shall not prevail against it. And I will give to thee the keys of the kingdom of heaven. And whatsoever thou shalt bind upon earth, it shall be bound also in heaven: and whatsoever thou shalt loose on earth, it shall be loosed also in heaven". Then he commanded his disciples, that they should tell no one that he was Jesus the Christ (Matthew 16:13-20; Mark 8:27-30; Luke 9:18-21).