He also told Peter that whatever he binds on Earth is bound in Heaven.
True. In Matthew 16 Jesus says this to Peter, and in Matthew 18 He repeats it to all the disciples. However, when given a wide open, golden opportunity to designate Rome as anything special, Jesus took the opposite tack:
John 4:
21 Jesus said to her, “Woman, believe Me, an hour is coming when neither in this mountain nor in Jerusalem will you worship the Father. 22 You worship what you do not know; we worship what we know, for salvation is from the Jews. 23 But an hour is coming, and now is, when the true worshipers will worship the Father in spirit and truth; for such people the Father seeks to be His worshipers. 24 God is spirit, and those who worship Him must worship in spirit and truth.”