I hope I haven't waited too long to respond.
I read the documents you linked to, at least a good portion of them but I disagree with some of the most basic statements. First the Eastern and Roman Church had a disagreement and split. They both are from the same place, they both have what they claim are a basis for their own legitimacy and I think I am unable to say which has the better argument. Since they have been for so long trying to see a way back to being one I suspect they are having the same difficulty as me.
Since the Coptic churches were really what they were from before 325 I am inclined that they have very much a good argument of being from the beginning of Christendom. The Apostle Mark started them during his lifetime. As it turns out Bishops from the Coptic Church authored most of what we now have as the Nicene Creed. So if anything they predate the “Roman” Catholic Church.
I have never claimed that Protestants have any claim to legitimacy as the original church or any authority to baptize or any other ordinance that Christ says we must have to enter Heaven.
As far as the Church remaining until The Savior returns, I think there are arguments that a reasonable person could infer that that argument is invalid, not that it is, just it could be reasonably argued. The Apostle Paul actually argues that there must be a “falling away first” before the Lord returns. I don't know when the falling away is to have happened or when it is to happen but anyone who proffers that the Church will stand in perfection till He returns does not have the best of sure footing.
“You are Peter, and upon this rock I will build my church”.
Peter went to Rome. Peter was the first Pope of the Catholic Church. All Christians faiths are off-shoots of the Catholic Church.