The Roman syncretism was probably one of the reasons Christianity supplanted paganism in the Empire.
And, correct me if I’m wrong, but wasn’t the persecution of Christians caused by their refusal to participate in and acknowledge the Imperial Cult, and not because of them acknowledging no other gods?
As I understand it, by not worshiping other gods they were jeopardizing the stability and security of the state and people. The crime was serious. The general population had a multitude of gods that they could worship as they wished. By pleasing all of the gods indiscriminately the state would theoretically be less likely to suffer the mischief of some otherwise neglected angry deity and everyone would benefit.
The Jews were given a special exemption from this, because of the great antiquity of their religion, which was recognized by the state as a valid true religion (which had a god that accepted worship only from it’s own chosen people, so to get that god’s support the state need those people to pray to it).
When the Christian sect was generally cast out of the synagogs the allegation made was that they were not real Jews, but a new sect of heretics and blasphemers whose prayers to a false god (and only a false god) were meaningless.
The state agreed and did not allow the new religion the exemption granted to the Jews.
The ‘test’ chosen was to make a gesture of worship to the emperor, probably because it was the only worship every citizen (except Jews) was legally bound to make (worshiping other gods was always an option and encouraged, but the choice of which ones was up to the individual).
But the crime charged against the Christians was actually atheism. They were accused of not believing in real gods, and worshipping a god that (being new) did not actually exist. This was considered highly offensive to the pantheon of gods, and the persecution of these people would put the state on the ‘right side’ of the argument and keep the empire in the various god’s good graces.
This is my interpretation of the situation, however someone else may have another take on it.