It is true that Christianity was growing exponentially within Rome during the 3rd and 4th centuries, and that Constantine was, in classic politician form, simply acknowledging events on the ground that had long been happening. Likewise Theodosius established Christianity as the state religion as Christians became a supermajority. The story that Constantine's conversion somehow enabled the spread of Christianity is a myth.

However, I have never heard anything about a Jewish family structure being an attraction of Christianity. If anything after the first century Christianity was much more a Greek-influenced religion.