This very old Protestant claim is no longer believed even by modern Protestant scholars, as the evidece that Peter went to Rome is historically incontrovertibe. Many of the early Church Fathers and historians wrote of Peter living and dying in Rome. Here is just one such proof of Saint Peter's residence in Rome:
"But since it be too long to enumerate in such a volume as this the successions of all the Churches, we shall confound all those who, in whatever manner, whether through self-satisfaction or vainglory, or through blindness and wicked opinion, assemble other than where it is proper, by point out here the successions of the bishops of the greatest and most ancient Church known to all, founded and organized at Rome by the two most glorious Apostles, Peter and Paul, that Church which has the tradition and faith which comes down to us after been announced to men by the Apostles"
Just to be clear on the subject, I'm not actually arguing herein against the possibility that Peter visited Rome (once, or more than once).
The main point of the Article is that according to the evidence of the Bible, he didn't spend much time in Rome (throughout the 40s, 50s, and 60s AD, he's sojourning in Jerusalem, Antioch, etc., but not Rome); and according to the evidence of Archaeology, he was not buried in Rome.