I see where you’re coming from here now.
Yes, much of what the apostles handed down orally through tradition was the Gospel message. Do we know for a fact that there was nothing else they taught orally? Not really. There are clues, though. Scripture itself attests to the fact that many of the actions, and presumably, teachings, of Christ were not written down anywhere (Jn 21:25). If it was important enough for John to write that fact down, it was probably important to somehow retain these extra-biblical teachings of Christ, and hand them on orally through the apostolic family. Other clues are the early Church Fathers, especially those directly linked with the some of the 12 while they were still living…like Polycarp and Clement. The verified writings of these, and many other, fathers of the Church, expound upon Scriptural writings and demonstrate or reflect a consistent doctrinal standard that is often not explicitly found within those same scriptural writings.
To answer your other concern…no, The Church is not the real Church of Christ because it says so, or because it’s old. It happens to be the only Church in the world which DOES claim to be the One True Apostolic Church, and that alone is a good reason to give due credence to it’s authenticity. But more than that, She happens to be the only Church whose leadership CAN be traced directly back, successively through ordination, to St Peter and the 11. No other church on earth can do that (save for the EOC which is in communion with The Church). Beyond that, the infallibility is there because Christ deemed it to be. The teaching of faith and morals needed to be protected from error, because humans can quickly mess things up, whether it be transmitting teachings orally, or by interpreting written words. Infallibility is a gift from the Holy Spirit that protects the Holy See when he is proclaiming a truth regarding faith and morals ex cathedra (from the chair of Peter). If you search this site for ‘infallibility’, you will see that it is an often misunderstood charism of The Church.
I would say that the prophets were infallible in much the same way, yes. They were guided by God to teach faith and morals to the people of the age of Messianic prophesy, much like The Church does now for the age of the risen Messiah.
God Bless