As we all should who seek real truth. It is ever continuous.
Actually, we see in Acts 20:30, a breakaway (false church) from the real church, promoting contrary doctrines and drawing away disciples from the real church. John spoke of it as well. I John 2:18,19. Paul also lightly touched on it in II Thess.2:2-4. We need to examine the Scripture's, the Word(s) of Christ, and the early teachings of the disciples to see what could possibly be contrary.
I wouldn't put too much stock in being sure one Is necessarily the real church by it's longevity. The little horn power in Daniel reigns for a time, times, and a dividing of time (1260 yrs.). And the beast from the sea in Revelation was to reign for 42 months (1260 yrs.) until it receives a wound (1798 a.d.), which would be healed (Mussolini 1940's). After Imp. Rome...the Restrainer, comes the RCC 538ad till 1798ad when she receives a deadly wound by General Berthier. She was restored (healed)to power by Mussolini. and continues today, till the end.
Besides, God woman (the true church) went into hiding for 1260 yrs. (12:14).
It's not a list of successors that determines if its God's true church, it the truth's the Scriptures speak about (gospel)that's important.
I wouldn't count on Ignatius as a reliable source. His supposed writings are highly disputed, being labeled antedated forgeries and interpolated throughout. The RCC doesn't deny it has used such strategies to maintain its power, when found out.
No one disputes there was a roman church, but the embellishments that prop it up as the ONLY are called into question.
Every believer (ecclesia) along the way, from Pentecost forward, are His church His building. They were, and are added daily, being built up.
Of course he did. The delivering of the gospel was a tedious task. But it was being published throughout. (Acts 6:7a; 10:37; 12:24; 13:49; 19:20). Blessings!