The first church certainly wasn't formed in Rome. The earliest
church was run by the Jerusalem Council of apostles, call it the
Jerusalem Church.
In fact, I suspect the church in Rome started later, after the
churches in Alexandra and Antioch were established. Correct me if
I'm wrong.
Furthermore, in the beginning, the church in Rome was not as
important as the ones in Alexandra and Antioch.
I think it is logical also. Most Roman rulers before Constantine
were unfriendly to Christians. You think the church in Rome would
have an easy life? How can it thrive under the foot of the
emperor?