John and the Synoptic Gospels
Christ Jesus did not carry the Gospels with Him while He was preaching in Israel 2000 years ago, since no Gospels were written yet.
So, it is unlikely that Peter and Paul preached in Rome with the Gospels at their hands. Instead, they must have preached like Christ Jesus even if the Gospels according to Luke, Mark, and Mathew had been completed at the time when Peter had joined Paul in Rome after the completion of the Acts by Luke.
Put simply, the Gospels according to Luke, Mark, and Mathew were not written for Paul and Peter. Then for whom these Gospels were written? To distribute them to Roman citizens?
Among Roman citizens who came to believe Peter and Paul and thus Christ Jesus, some active followers must have emerged. And these Roman followers must have preached to other citizens in lieu of Peter and Paul. In this way, Christianity began to spread among Roman citizens.
But to preach to other citizens, these Roman followers needed some sorts of guidelines. The Old Testament was however useless for them. Accordingly, they probably asked Peter and Paul to present some Greek material that would help them preach in Rome. For this reason, Paul asked Luke and Peter asked Mark to write Greek versions of the Gospel based on the Hebrew/Aramaic source document Paul had and another source Mark had made by hearing words of Christ Jesus from Peter on their journey before they had reached Rome.
So, the Gospels according to Luke and Mark must have been written in order to teach Roman leaders of followers how Christ Jesus had preached but not to be distributed among the general public of Rome. These Roman leaders learnt the mission of Christ Jesus from these Gospels, and based on them, they preached to other Roman citizens. Probably they referred to the words and the episodes of Christ Jesus written in these Gospels. But they did not use the Gospels in a manner that current priests use the Bible in their Mass. These Gospels must have been a kind of textbooks for Roman Christian leaders since Paul and Peter started their preaching in Rome. Accordingly, the Gospels according to Luke, Mark, and Mathew are called synoptic Gospels. In other words, they were not written to disseminate the theology of Christianity but as reference material to support preaching by Roman Christian leaders.
However, after the death of Peter and Paul and the Jewish-Roman War, the situation changed. There was no more the temple in Jerusalem where Christ Jesus preached and died. The very early Christianity lost a tie to Judaism and could not rely on Judaism as the the underlying religion that offered basic theology to the very early Christianity.
Therefor John had to write his version of the Gospel with focus on theology. The Gospel according to John was intended to give religious authority to the very early Christianity by replacing Judaism symbolized by the temple in Jerusalem. Yet, he had to rely on the source documents having been brought to Rome by Paul and Peter with Mark. That is why the Gospel according to John depicts almost similar episodes and words of Christ Jesus to those in the synoptic Gospels but presents a very different view on God and Christ Jesus from the synoptic Gospels.
So, the Gospels according to Luke, Mark, and Mathew were written in the era of Paul and Peter when the very early Christianity had still a tie with Judaism. But with the Gospel according to John, the early Christianity started leaving the tie with Judaism completely behind.
So, from older days, scholars have made distinction between John and the synoptic Gospels. But they do not seem to clearly understand the reason. It is because they have not wondered how the Gospel had been preached in Rome by Peter, Paul, and Roman leading followers without the Bible we know today.
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Joh 2:13 And the Jews' passover was at hand, and Jesus went up to Jerusalem.