It’s possible that White’s sources are correct, but that possibility does not destroy the reality of the papacy.
We know that the first four popes were all contemporaries (as Clement was the last bishop of Rome to have known Peter personally). Further, even in the oldest lists of the popes, there seems to be confusion as to the exact order in which Linus, Cletus/Anacletus, and Clement assumed the office.
Furthermore, the New Testament seems to indicate that the titles of “presbyter” and “bishop” were not quite as distinguished as they later became. The structure of “single bishop with presbyters assisting him and deacons assisting them” likely evolved after the apostles had passed from the scene, since as long as there are living apostles the precise nature of apostolic succession isn’t a pressing matter. Still, that structure must have developed quite early, concurrent with the writing of the later New Testament books, as it seems to have been pretty firmly established by the time Ignatius wrote his letters in 107. Indeed, we can sort of see this development in the NT itself, as in some places the titles of presbyter and bishop seem to be used interchangeably, while in I Timothy separate qualifications are listed for the two offices.
So I’d say it’s at least possible that the men we now recognize as the 2nd-4th bishops of Rome did not succeed each other in a nice linear progression as we would see today, but served as “co-bishops” of a sort. It’s reasonable to presume that there would have been a “senior bishop” among the trio, as Peter undoubtedly was when all four were alive, but we don’t know enough historically about any of these men’s lives to say for sure.
The important thing is that, even as early as Clement’s time, the Church at Rome was recognized as the “court of appeal” for other Christian communities. Once it became the custom to have a single bishop over each city or region (whenever that occurred), that authority was logically personified in whatever man held the bishop’s seat in Rome.
Since we don’t have any records of Linus and Cletus disputing over some key teaching (nor, indeed, records of them doing much of anything at all), the fact that we can’t quite be sure exactly when each one was “the Pope” isn’t a terribly devastating thing.