Slop.

You raise a fair point; my earlier statement was misleading. The masses in general were not yet Christian, but the Christian minority showed a remarkable fervour that the rich no doubt noticed and were eager to exploit:

In a way that the awesome power-deities of the pagan pantheon could not, the all-powerful and supremely benevolent Christian god offered ‘a heart in a heartless world’ that had strong appeal to the oppressed of the Roman Empire.

[…]

The exploitation and oppression of the Roman Empire meant misery for millions, but the violence of the state usually prevented effective resistance. This was the contradiction that allowed the Christian Church to grow and grow.

Recruiting among slaves, women, and the poor, the Church was viewed with grave suspicion, and was repeatedly battered by repression. It [did] not work. The men and women set on fire, eaten alive by animals, or nailed onto wooden crosses to die provided the Early Church with a roll call of martyrs as impressive as any in history.

By the early 4th century CE, the Church had become the most powerful ideological apparatus in the Mediterranean world, with a complete underground network of priests, congregations, and meeting-places extending across the Empire.

Many army officers, government officials, and wealthy landowners had already become Christians. In 312 CE, the Emperor Constantine the Great decided to adopt Christianity himself, to legalise the religion, and to make the state the protector and patron of the Church. Before the century was out, his successor, Theodosius the Great, would make paganism illegal and hand over all temple estates to the Church.

(Emphasis added. Source.)

The ability for a minority to seize state power was a structural defect of the Roman Empire:

The army became more and more the master of the republic. As the mercenary soldiery increased, the fighting capacity of the Roman citizens fell; or rather, the decline of their fighting capacity conditioned the growth of the mercenary soldiery. All the elements of the people that were capable of fighting were in the army; the part of the people outside of it kept losing both its ability and its desire to bear arms.

[…]

The more non-Romans there were in the army and the more the aristocratic officers were replaced by career men, the more willing the army was to sell itself to the highest bidder and make him the ruler of Rome.

In this way the foundations were laid for Caesarism, by having the richest man in Rome buy up the republic by purchasing its political power. It was also the basis for having a successful general with an army at his back try to make himself the richest man of Rome; the simplest way to do this was to expropriate his opponents and confiscate their property.

(Source and more here.)

Sorry to overwhelm you with text, but I hope that this helps regardless.