Pliny's letter to Trajan on the topic: "What do I do about the Christians?" I find very revealing -- (and Trajan's response, too!) Trajan makes his opinion very, very clear. You cannot prosecute someone because you suspect he is a Christian, you prosecute him for the crime (arson, murder) you suspect he committed.
The Romans had a lot of misunderstandings about Christianity - initially, it was thought to be a weird sect that practiced cannibalism -- the "blood of Christ" was misunderstood to mean drinking of human blood, etc.
The group that had the worst wrap from the Romans during this period were the Jews. Nero, Vespasian, Titus Domitian were all horrible to the Jews and to Jerusalem. Trajan was a breath of fresh air, then Hadrian returns with gusto. IMHO, the Jews had it far worse.
Initially, many early Christian congregations held two "services" a week. Sunday was a worship service open to all. Wednesday was a closed service to those with commitments to Christ. History does not really tell us what went on in the closed services, but the Bishop of Rome strongly suggested that congregations either stop the Wednesday services or open them up to all. Some historians think this "secrecy" helped fuel the misunderstanding you have described.
You know quite a bit about the Roman Empire and the early church. How did you get to know so much. I am impressed!