You're confusing (although you seem to recognize this) the start of the Imperial Church in the Empire with Christianity from the start. The first Pope was Peter, then Linus, etc. Basically, just because it wasn't an official, organized institution that we see having arose after the Battle of Milvian Bridge, doesn't mean this Church is distanced in terms of heritage from the early Christianity. Where did you read the Catholic Church 'started' with Constantine?