Another good analysis, I think. By tedious, I meant that polytheism was no longer relevant to their society. As knowledge grew, it was tough to believe in those oh-so-human deities. So, we're both right, in a way.
Christianity was tailor-made for Rome, and of course, for the rest of Europe, as Rome went a-conquering. A perfect fit.
Actually, the Roman Empire had essentially reached its greatest extent in Europe at about the same time that the apostle Paul arrived at Rome. The growth of Christianity was not coextensive with the political power of Rome; many of the barbarians outside the borders were predominantly Christian. The Romans did not engage in missionary conquest; this is a modern fantasy.
Christianity was actually very poorly suited to Rome. Roman greatness had been based on two ideas: one, that the idea of Rome itself was a deity (expressed as Quirinus, the god of the Roman race, the genius of the City of Rome, Jupiter Capitolinus, etc.), and two, that each individual essentially had a duty to attain as much early glory as possible (expressed in political life as the cursus honorum, or obligation to climb the political ladder, or in military life as the idea of taking enemy prizes to dedicate back home to the gods).
Christianity was antithetical to these ideals. First, it expressed that all men were equal - that it was meaningless to be a Roman citizen, or a slave, or a patrician, or any other earthly status. Second, it expressed that earthly glory was meaningless - being an emperor was pointless if your immortal soul was imperiled. The vivid contrasts between the Christian ideals and the old Roman ideals produced violent conflict in the first centuries of the Christian era, and (according to some) eventually contributed to the collapse of the Western Roman Empire.