most of the writers of the new testament never lived with Christ physically.
So what? To use that fact as support for disbelieving Scripture is to deny the work of the Holy Spirit.
We have Matthew, Mark, Luke, John, Luke, Paul, Peter, James, Jude, John.
Of those writers, Luke definitely did not live with Jesus, but his words are from the records of the Christians that he compiled under their guidance while in Jerusalem.
Paul studied under Gamaliel and was a recognized leader to the extent that he received letters from the High Priest to capture Christians. He was the ring-leader of those who stoned Stephen. Since this occurred some time before the death of Herod Agrippa I (44 AD) mentioned in Acts 12, a mere ~11 years after the founding of the Church on Pentecost, we have Paul in Jerusalem at the time of Jesus or immediately thereafter. This is certain because Paul had been sent back to Tarsus by his friend Barnabus and then was searched out for help in Antioch by Barnabus. Paul in Galatians mentions 3(?) years being taught by the Lord in the desert.
Therefore, in that 11 year period we have to account for Paul's conversion, his 3 year training period, his return to Tarsus, and his time in Antioch.
Paul knew much of what he knew from having lived in Jerusalem. Also, don't forget that Paul's sister lived in Jerusalem as did his nephew (who warned him of a plot.) This suggests immediate longstanding family presence in Jerusalem. As a student of Gamaliel and as a man with immediate family in Jerusalem, it is fairly safe to suggest that Paul had extensive contacts in Jerusalem.
Paul, of course, was also a witness to the Resurrected Christ, and no Apostle disputed that. Instead, Acts, Peter, and Galatians portray that fact as fully accepted by the Apostles.
Therefore, of the list of NT authors, there is only 1 who was not present during Jesus' ministry or the early formation of the church. There is probably only one who was not "physically" present at the time of Jesus' ministry.