Kinda? Paul was definitely a huge part of spreading Christianity, but we also know many important churches had no association with him at all. He writes to the Roman church as someone who's never visited them. We never hear of him visiting Alexandria, another very important early Christian site. One of the biggest names at the beginning of the second century is in France (St Irenaeus of Lyon). Paul is a very important part of the early movement, for sure, but the other disciples and their disciples were too.

As far as first hand writings: Paul's letters are the earliest NT docs (1 Thessalonians is early 50s we think), that's true. But a few of the other works aren't much later. For example, Mark's gospel is not that much later, around 60-70CE. Mark very easily could have interviewed eye witnesses or descendants of them, though this point is heavily contested admittedly. But yes, the rest are later. Matthew and John were written later still in the 80s, Revelation and John's Gospel sometime in thr 90s and Acts was probably written around the turn of the century. The pastoral letters may be as late as 125CE. So the foundation of the theory seems okay, but I don't know of any historian or scholar who argued the above theory personally.