Does Paul Refer to Ongoing Authoritative Extra-Biblical Tradition in 2 Timothy?

What constitutes the ongoing infallible authority for the church? Roman Catholic apologists say it consists of Scripture + Tradition +  a Church magisterium (often referred to as"three-legged stool" model of authority).
Protestant  limit the infallible rule of faith to sacred Scripture. Two key passages relied on by Protestants are as follows:

2 Timothy 3:14-17
14 You, however, continue in the things you have learned and become convinced of, knowing from whom you have learned them, 15 and that from childhood you have known the sacred writings which are able to give you the wisdom that leads to salvation through faith which is in Christ Jesus. 16 All Scripture is inspired by God and profitable for teaching, for reproof, for correction, for training in righteousness; 17 so that the man of God may be adequate, equipped for every good work.

Romans 16:25-27
25 Now to Him who is able to establish you according to my gospel and the preaching of Jesus Christ, according to the revelation of the mystery which has been kept secret for long ages past, 26 but now is manifested, and by the Scriptures of the prophets, according to the commandment of the eternal God, has been made known to all the nations, leading to obedience of faith; 27 to the only wise God, through Jesus Christ, be the glory forever. Amen.

In his book, Holy Scripture: The Ground and Pillar of Our Faith, Volume I: A Biblical Defense of the Reformation Principle of Sola Scriptura, David King has commented,

Sola Scriptura functions as the authoritative norm for the people of God, and therefore stands as the only existing source of the deposit of faith that special revelation has disclosed. This is one reason why Paul wrote as he did in his closing remarks in Romans. He spoke of 'the revelation of the mystery kept secret since the world began but now has been made manifest, and by the prophetic Scriptures has been made known to all nations, according to the commandment of the everlasting God, for obedience to the faith' (Rom 16:25-26). According to the testimony of the Apostle Paul, Scripture has manifested this mystery, and this mystery is authoritatively interpreted by the epistles of the New Testament.

Scripture is therefore authoritatively specified (via apostolic sanction) as the God-ordained means for the manifestation of this mystery, the purpose being 'for obedience to the faith.' If unwritten tradition was to be regarded as a reliable means and/or source for the preservation of binding revelation beyond the time of the apostles, and intended to function perpetually as an authoritative norm alongside Scripture, why did Paul fail to mention such a concept when speaking of 'the revelation of the mystery kept secret since the world began?' (p.44)

Some Roman apologists claim Paul did not omit ongoling oral tradition as an authoritative norm alongside Scripture that was to function as an infallible rule of faith for the church. The most typical verse cited is 2 Thessalonians 2:14-15. I've dealt with that argument elsewhere, as have much abler writers (see David King's treatment on pages 119-121). I recently saw another proof offered for the Roman "three-legged stool" paradigm coming from one on the chief chapters Protestants rely on for sola scriptura, 2 Timothy 3.

2 Tim. 3:10; 14
10 Now you followed my teaching, conduct, purpose, faith, patience, love, perseverance, 11 persecutions, and sufferings, such as happened to me at Antioch, at Iconium and at Lystra; what persecutions I endured, and out of them all the Lord rescued me! 12 Indeed, all who desire to live godly in Christ Jesus will be persecuted. 13 But evil men and impostors will proceed from bad to worse, deceiving and being deceived. 14 You, however, continue in the things you have learned and become convinced of, knowing from whom you have learned them,

Here Paul is said to be teaching an ongoing infallible extra-biblical tradition as a norm along side of sacred Scripture. But, as David King has pointed out above, saying such exists and proving it are two very different things. David rightly asks of any Roman apologist claiming such:

In the end, the matter of authoritative revelation boils down to a question of preservation. Regardless of the assertions of Roman apologists, when the dust settles, there is one question they cannot answer. Can you name one oral, extrabiblical tradition, demonstratively traceable to the apostolic age, which is necessary for the faith and practice of the Church of Jesus Christ? No verifiable example has been or can be offered. We affirm with the Apostle Paul, 'that according to the Way which they call a sect (or heresy), so I worship the God of my fathers, believing all things which are written in the Law and in the Prophets' (Acts 24:14).

Certainly we can agree with Rome's apologists that previous to the completion of the New Testament and the deaths of the apostles and New Testament writers, the Word of God had an oral dimension. We can certainly agree that Paul taught Timothy in an oral nature, and that teaching was the word of God (2 Timothy 1:13-14, What you heard from me, keep as the pattern of sound teaching, with faith and love in Christ Jesus. Guard the good deposit that was entrusted to you—guard it with the help of the Holy Spirit who lives in us). David King though rightly states that "In the end, the matter of authoritative revelation boils down to a question of preservation." It certainly does. If extra-biblical infallible oral revelation intended for the Church exists,  where is it, and what is it? What did Paul teach Timothy other than what we have in the Bible, and where is it? As King said above, "If unwritten tradition was to be regarded as a reliable means and/or source for the preservation of binding revelation beyond the time of the apostles, and intended to function perpetually as an authoritative norm alongside Scripture, why did Paul fail to mention such a concept when speaking of 'the revelation of the mystery kept secret since the world began?" As the old TV commercial goes, the question Rome's apologists need answer is where's the beef?

Other Spurious Examples of Infallible Extra-Biblical Tradition
Along with 2 Tim. 3:10;14, a Roman apologist may put forth Jannes and Jambres in 2 Timothy 3:8-9 as an example of infallible extra-biblical tradition:

8 Just as Jannes and Jambres opposed Moses, so these men also oppose the truth, men of depraved mind, rejected in regard to the faith. 9 But they will not make further progress; for their folly will be obvious to all, just as Jannes’s and Jambres’s folly was also.

David King points out,

Roman apologists insist that the names of Jannes and Jambres would be unknown to us had they not been preserved by means of oral tradition. But this does not prove extrabiblical tradition, only inscripturated tradition. More importantly, the knowledge or ignorance of these names has no bearing on one's eternal destiny. This is not a matter of de fide even within the Church of Rome. Additionally, we draw from the witness of the Church father, Chrysostom, who said, 'Who are these [i.e., Jannes and Jambres]? The magicians in the time of Moses. But how is it their names are nowhere else introduced? Either they were handed down by tradition, or it is probable that Paul knew them by inspiration.' Theodore of Mopsuestia (350-428) wrote that, 'A great deal of foolishness has been written about how Paul could have known the names of these two men who resisted Moses.' Theodoret of Cyrrhus (393-466) stated that, 'Paul takes the story of Jannes and Jambres not from holy Scripture but from unwritten Jewish tradition.'Instances of Alleged Support for Extra-Biblical Tradition. These examples demonstrate that there was a diversity of opinion among the early Church fathers themselves, thereby proving that there was no objective tradition current in their time. Perhaps, Theodore of Mopsuestia gave the best summation of speculative arguments over Paul's source when he spoke of the 'foolishness' that had been written about it. (p. 123).

Another popular argument using 2 Timothy against sola scriptura comes from 2 Timothy 2:1-2

You then, my son, be strong in the grace that is in Christ Jesus. 2 And the things you have heard me say in the presence of many witnesses entrust to reliable people who will also be qualified to teach others.

David King points out,

While the defenders of Rome quote this passage in support of extra-biblical traditions, we know that this was no secret deposit or esoteric oral tradition. Tertullian wrote:

What is this deposit? Is it so secret as to be supposed to characterize a new doctrine? or is it a part of that charge of which he says, 'This charge I commit unto thee, son Timothy?' and also of that precept of which he says, 'I charge thee in the sight of God, who quickeneth all things, and before Jesus Christ who witnessed a good confession under Pontius Pilate, that thou keep this commandment?' Now, what is (this) commandment and what is (this) charge? From the preceding and the succeeding contexts, it will be manifest that there is no mysterious hint darkly suggested in this expression about (some) far-fetched doctrine, but that a warning is rather given against receiving any other (doctrine) than that which Timothy had heard from himself, as I take it publicly: 'Before many witnesses' is his phrase. Now, if they refuse to allow that the church is meant by these 'many witnesses,' it matters nothing, since nothing could have been secret which was produced 'before many witnesses.' Nor, again, must the circumstance of his having wished him to 'commit these things to faithful men, who should be able to teach others also,' be construed into a proof of there being some occult gospel. For, when he says 'these things,' he refers to the things of which he is writing at the moment. In reference, however, to occult subjects, he would have called them, as being absent, those things, not these things, to one who had a joint knowledge of them with himself. (emphasis mine).

According to Tertullian, Paul committed to 'faithful men,' who in turn passed on this deposit of truth, the things of which 'he is writing at the moment.' It had nothing to do with extrabiblical traditions. It was a succession of apostolic teaching, not a succession of men.

Among the things Paul conveyed, in this epistle, was his own instruction regarding the sufficiency of holy Scripture. Thus, this 'proof' text does not support the Roman contention regarding an oral deposit of doctrine passed down through successive generations in the Church. (pp.122-123).

Conclusion
These few examples demonstrate that David King's book, Holy Scripture: The Ground and Pillar of Our Faith, Volume I: A Biblical Defense of the Reformation Principle of Sola Scriptura, is of immense value in dealing with Roman apologists.  To this date, no Roman Catholic apologist has offered any sort of meaningful comprehensive counter-response. Those that have made attempts typically start, but give up. I can think of at least two Roman Catholic apologists that guaranteed full rebuttals, but quickly gave up. Some actually make arguments against King's book, not even realizing he's already answered their argument. For instance, one Roman apologist mentions Jannes and Jambres in 2 Timothy 3:8-9 as an example of infallible extra-biblical tradition that refutes King, apparently not even realizing King's book included a section on Jannes and Jambres.