It is impossible for you or me to appeal to any oral teachings of Timothy, or Paul, or Christ. But we can, and MUST appeal to what we have preserved for us in Divinely Inspired, Holy WRITINGS. Just as Christ and the Apostles did in their day and rejected the “traditions” that did not conform to what was already written. In fact, it’s how they tested the traditions of their day.
What happened at Pentecost? The Church is God-breathed from the beginning (not just the Scriptures, which, in fact, come later and must be determined by the God-given authority of the Church) and the effects of this God-breathing in the Church are far greater than the effect in Scripture which is inanimate. The Holy Spirit descends on the gathering and makes it into a new supernatural creation, the Body of Christ, which is a living Temple, God dwelling within His People as an organism, a new humanity living by the Life that is in the Head, i.e., the Divine Life of the Holy Spirit and His communion between the Father and the Son; individuals are living stones of this Temple.
This Church, which before Pentecost was like the clay with which God formed Adam, is now breathed into by Christ with the Breath of the Holy Spirit, the way Adam was breathed into by God to make him a living being. The Church is not just a society of people that God is working in, but has been made into the Bride able to marry Christ. It is a supernatural creation descending from Heaven onto the gathered disciples ensouling them with divinity, the Divinity of the Trinitarian indwelling.
If Scripture is the written words of God, the Church is God’s living Body to which, for and within which His written words were communicated after His oral words were communicated; but more than this, if the Scriptures are God’s word(s), the Sacraments are God’s actions, God-breathed through and through, and the Eucharist His Eternal Wedding Feast already begun, where the nuptials are exchanged, where the Covenant is sealed with His now divinized Blood and Risen Flesh which becomes the food of the banquet.
This is not so much deduced from Scripture as intuited from it by juxtaposing all the meanings of the Church in the New Testament (and presaged in the Old), of presence; of images the NT uses to convey the Kingdom of God already begun. Infallibility of the Church, then, would be a corollary that follows from this spiritual insight. More important is the awareness that despite being a vessel of clay, the Church holds within that clay of its wounded humanity not mere water, but the Heavenly Wine of Divinity, the Libation of the Wedding Feast of the Lamb Who has transformed the mere water of the human spirit by breathing the Breath of Divine Love, the Third Person from the Father, into these human vessels and filling them with the wine of His Life-Blood, which is far better wine kept for last.
Logic won’t help you here, but intuition will; or, rather, logic is only going to work once the insight is attained. So consider the scene of Pentecost in depth with all the other images of the Church of the NT, and see that event manifesting the Church being God-breathed. This is the NT understanding of the Church.
And there in that scene at Pentecost we see St. Peter. He is not conferred authority by the other eleven to speak for the Twelve when preaching the first kerygma and ordering the converts to be baptized so as to receive the Divine Birth. He is impelled by the Holy Spirit, having already been commissioned by Christ both before and after the Resurrection. IOW, at Pentecost it is not as if St. Peter just happens to be so inspired and the others recognize this and don’t interfere with the Spirit’s momentary choice of instrument, as if there was no previous history about Christ and Peter. It is the same Christ Who sends the Holy Spirit inspiring St. Peter at Pentecost Who gave the commission to him before the Ascension as well as the prophecy of that event before His death.
And at the Council of Jerusalem, St. Peter takes the initiative for the doctrinal judgment (while St. James speaks up on the disciplinary consequences that will affect his sphere of influence). St. Peter’s assertion of judging the issue doctrinally (following upon the previous precedent of baptizing Cornelius) is an assumed authority for speaking for the rest. My point is that there is already in place a principle of visible order coming from Divine establishment and not left for merely ecclesiastical or civil arrangement to grant.
The New Testament already has an identifiable leader of the Twelve who can on his own speak for them deciding doctrinal issues (which were not raised by him) and this does not contradict another of the Twelve, James, from exercising local initiative and authority over his allotted community and contributing to the larger picture.