No, it wouldn't, first of all, because they are unable to transmit the apostolic mission; and secondly, this is not a refutation of a heresy at all. Have you read St. Alphonsus' "Refutation of All Heresies". According to Sacred Theology, there is a specific way, or at any rate, many specific ways, to refute each and every heresy, whether it is Monophysitism, Monothelitism, Arianism, Nestorianism, Gallicanism etc.

So how will you refute, from some clear premise, the heresy of Old Catholicism, and prove and establish that it is heretical to hold? Just as an e.g. here is one of the ways in His Holiness Pope Bl. Pius IX, refuted it, from the Indefectibility and Universality of the Church: "Incredibly, they boldly affirm that the Roman Pontiff and all the bishops, the priests and the people conjoined with him in the unity of faith and communion fell into heresy when they approved and professed the definitions of the Ecuмenical Vatican Council. Therefore they deny also the indefectibility of the Church and blasphemously declare that it has perished throughout the world and that its visible Head and the bishops have erred. They assert the necessity of restoring a legitimate episcopacy in the person of their pseudo-bishop, who has entered not by the gate but from elsewhere like a thief or robber and calls the damnation of Christ upon his head ...Moved by your voices and your false opinions, She asked of God that He announce to Her the length of Her days and She found that God said ‘Behold I am with you all days even to the consummation of the world.’ Here you will say: He spoke about us; we are as we will be until the end of the world. Christ Himself is asked; He says ‘and this gospel will be preached in the whole world, in testimony to all nations, and then will come the end.’ Therefore the Church will be among all nations until the end of the world. Let heretics perish as they are, and let them find that they become what they are not.”[8] https://www.papalencyclicals.net/pius09/p9etsimu.htm

Of course they were wrong, and of course it did continue in the Popes and their Successors.

No, it wouldn't continue either way. That is heretical to say. If it is even possible that Vatican I was not a legitimate Council, then it would follow that Papal Infallibility is not a dogma known with certainty of faith. That's why theologians explain such things are dogmatic facts. Similarly, we know a vacancy going back to 1870 is heretical.

Partially agreed. The SSPX holds that after Summorum Pontificuм in 2007, which corrected an injustice and admitted that all Priests were free to offer the Traditional Mass; after Universae Ecclesiae some 4 years later, and some Roman Theologians admitting that the Council is non-infallible, and that certain points can be respectfully questioned etc more recently, there is no reason for any Traditional Bishop not to go to Rome, receive recognition from the Pope, and continue a Traditional Apostolate in fully normal relations with Rome. But that's the basis of the SSPX-Resistance dispute and so, if you'd like to discuss it, we can discuss it in that sub-forum.

God Bless.

History confirms that the state of necessity extended not only the duties of bishops, but also their power of jurisdiction. Dom Grea whose attachment to the pope is above all suspicion testifies (De l’Eglise et de sa divine consitution, vol. I) that not only at the beginning of Christianity did the "necessity of the Church and the Gospel" demand that the power of the episcopal order be exercised in all its fullness without jurisdictional limitations, but that in successive ages extraordinary circuмstances required" even more exceptional and more extraordinary manifestations" of episcopal power (ibid., p.218) in order "to apply a remedy to the current necessity of the Christian people" (ibid. and ƒƒ.), for whom there was no hope of aid on the part of the legitimate pastors nor from the Pope. In such circuмstances, in which the common good of the Church is also at stake, the jurisdictional limitations vanish and "that which is universal" in episcopal power "comes directly to the aid of souls" (ibid., p.218):

There is no doubt that the Church of Vatican 2 has taught errors. But that these errors were taught in a manner so as to engage infallibility, that remains to be seen.

Among the prerogatives conferred on His Church by Christ is the gift of indefectibility. By this term is signified, not merely that the Church will persist to the end of time, but further, that it will preserve unimpaired its essential characteristics. The Church can never undergo any constitutional change which will make it, as a social organism, something different from what it was originally. It can never become corrupt in faith or in morals; nor can it ever lose the Apostolic hierarchy, or the sacraments through which Christ communicates grace to men. The gift of indefectibility is expressly promised to the Church by Christ, in the words in which He declares that the gates of hell shall not prevail against it. It is manifest that, could the storms which the Church encounters so shake it as to alter its essential characteristics and make it other than Christ intended it to be, the gates of hell, i.e. the powers of evil, would have prevailed. It is clear, too, that could the Church suffer substantial change, it would no longer be an instrument capable of accomplishing the work for which God called it in to being. He established it that it might be to all men the school of holiness. This it would cease to be if ever it could set up a false and corrupt moral standard. He established it to proclaim His revelation to the world, and charged it to warn all men that unless they accepted that message they must perish everlastingly.

In asserting that the Church of Christ is visible, we signify, first, that as a society it will at all times be conspicuous and public, and second, that it will ever be recognizable among other bodies as the Church of Christ. These two aspects of visibility are termed respectively "material" and "formal" visibility by Catholic theologians. The material visibility of the Church involves no more than that it must ever be a public, not a private profession; a society manifest to the world, not a body whose members are bound by some secret tie. Formal visibility is more than this. It implies that in all ages the true Church of Christ will be easily recognizable for that which it is, viz. as the Divine society of the Son of God, the means of salvation offered by God to men; that it possesses certain attributes which so evidently postulate a Divine origin that all who see it must know it comes from God.

There is no doubt that the Church of Vatican 2 has taught errors. But that these errors were taught in a manner so as to engage infallibility, that remains to be seen.

Bumped for Liarslaus. Liarslaus of course conveniently ignored answering this; because it destroys his heretical Apostolicity-Dogma denying heresies and schismatic mentalty.

Oh please. What is your explanation of the Vatican I text on Shepherds and Teachers? You can't answer it so you didn't.

Major Premise: Vatican I says there will be Shepherds and Teachers in Christ's Church until the end of time.