My argument is and always has been....the Twelve were not called to the Gentiles....and in fact were specifically told to stay away from them. In Acts 10 we see Peter hesitantly obeying the Holy Spirit and going to see Cornelius. All were astonished at the outcome of this visit and some Apostles even criticized Peter [Acts 11:2-3].
During this time Paul is not mentioned but has already experienced his conversion [Acts 9]. It had been announced immediately to Ananias [Acts 9:15] that Paul would be the one to carry the name of Jesus to the Gentiles. Can you imagine the shock of receiving this information....by the disciples? Paul then spends some time in Damascus and Jerusalem but has to flee for his life as the Jews are plotting to kill him [Acts 9:19-31] and is finally sent home to Tarsus by the disciples. Most of them (Christians) still did not trust him (verse 26).
Enter Peter and the Holy Spirit's mission to Cornelius.
Peter had been respected [Acts 5:15] and the Church grew quickly [Acts 2:40] but something had been brewing [Acts 6:1]. Division entered the early church between the Hebrew speaking Jews and the Greek speaking Jews.[Acts 6:8-10] The Greek Jews (Alexandrian) had turned on Stephen [Acts 6:8-10] and had attempted to kill Paul [Acts 9:29].
Now we have Paul pronouncing his intent to evangelize the Gentiles, and because of his prior reputation is it a wonder anyone in the Church trusted him? The Church was already being weakened because of the internal strife and now they had to deal with Gentiles....also?
This is when the Holy Spirit decided to get Peter involved....as a bridge between the Jews and Gentiles. Paul was evidently having great difficulty with his ministry and needed the support of the Jews (non Alexandrian). When the Holy Spirit showed Peter that no man was considered unclean the word traveled quickly. This had to help Paul in his ministry...but the original Eleven...and Matthias still had their own ministry and it did not include the Gentiles. This was left to Paul and hundreds of others.
We even notice....years later, Paul criticizing Peter for his continued Anti-Gentile stance [Galatians 2:11-14] and hypocrisy. This is biblical proof that Peter did not carry on a gentile ministry....particularly....to Rome!
Yes, Peter did visit Cornelius, a Gentile....a man of God, but his mission, along with the Eleven, was to the House of Israel. His letter is addressed to them [1 Peter 1:1-2] and other Apostles also address them specifically. [James 1:1]
As a Jew he would always have difficulty with gentiles. The animosity between gentiles and Jews was enormous of this period. Even Peter could have struggled with those prejudices, that does not mean that he was unable or did not rise about it to preach to them, as exemplified by Cornelius.