Writing in 1895, Fr. Sylvester Hunter, in Outlines of Dogmatic Theology, gives a simple refutation of the schism of sedevacantism: "First, then, the Church is infallible when she declares what person holds the office of Pope; for if the person of the Pope were uncertain, it would be uncertain what Bishops were in communion with the Pope; but according to the Catholic faith, as will be proved hereafter, communion with the Pope is a condition for the exercise of the function of teaching by the body of Bishops ...

This argument is in substance the same as applies to other cases of dogmatic facts. Also, it affords an answer to a much vaunted objection to the claims of the Catholic Church, put forward by writers who think that they find proof in history that the election of a certain Pope was simoniacal and invalid, and that the successor was elected by Cardinals who owed their own appointment to the simoniacal intruder; from which it is gathered that the Papacy has been vacant ever since that time. A volume might be occupied if we attempted to expose all the frailness of the argument which is supposed to lead to this startling conclusion; but it is enough to say that if the Bishops agree in recognizing a certain man as Pope, they are certainly right, for otherwise the body of the Bishops would be separated from their head, and the Divine constitution of the Church would be ruined."

Therefore, anyone who denies that a Pope universally recognized by the Bishops of the Teaching Church is truly Pope, necessarily denies the Indefectibility of the Church - he claims the Body of the Bishops has totally been severed from their Head, so that he ruins, by his false opinion, the Divine Constitution of the Church. 

Question for SVs: (1) How many of the more than 5000 Bishops in the Catholic Church today can you show have not accepted the last 6 Popes? You would probably need to show at least 500 have not (10%), yet it is unlikely anyone can show that even 50 (1%) have not.

Next, Msgr. Van Noort, writing in the time of Pope Ven. Pius XII, says that this Unanimous Acceptance of the Church Teaching is an infallible declaration of the Ordinary and Universal Magisterium (just like Pope Pius XII himself had said about the Universal Episcopate's Unanimous Acceptance of the Doctrine of the Assumption, even before it was dogmatically defined. "For — skipping the question of how it begins to be proven infallibly for the first time that this individual was legitimately elected to take St. Peter’s place — when someone has been constantly acting as Pope and has theoretically and practically been recognized as such by the bishops and by the universal Church, it is clear that the ordinary and universal magisterium is giving an utterly clear-cut witness to the legitimacy of his succession”

Second Question: (2) Where are the Bishops who comprise the Teaching Church today, per 60 year SVism? Are there any such Bishops at all, granted that Bishops can only be made members of the Teaching Church and Episcopal College through the Pope? But even if you want to claim all the Bishops appointed by Popes John XXIII to Pope Francis (who were not Popes in your opinion) are the Bishops of the Teaching Church, their Unanimous Acceptance of the current Popes, including and up to Pope Benedict XVI and Pope Francis, would prove that they are currently Popes. 

In order for someone to try to undercut this unanimous acceptance in future, he or she would need to convince the Bishops to stop recognizing the Pope. Has any SV done that? If not, the Catholic Church, who "is infallible when she declares what person holds the office of Pope" has infallibly judged.

Archbishop Lefebvre probably wavered a little on the question. Most of the time, His Grace was firmly against SVism. At certain times, he was unsure, true; yet, at other times, +ABL himself clearly expressed the idea, for e.g. on March 9 1980: 

"Archbishop Lefebvre's attitude was made very clear in a letter to the Sovereign Pontiff dated 8 March 1980. It reads as follows:

Quote   Most Holy Father, To put an end to some rumors which are now spreading both in Rome and certain traditionalist circles in Europe, and even in America, concerning my attitude and my way of thinking with respect to the Pope, the Council, and the Novus Ordo Mass, and fearing lest these rumors should reach Your Holiness, I may make so bold as to reaffirm my consistent position. 1. I have no reservation whatsoever concerning the legitimacy and validity of your election, and consequently I cannot tolerate there not being addressed to God the prayers prescribed by Holy Church for Your Holiness. I have already had to act with severity, and continue to do so, with regard to some seminarians and priests who have allowed themselves to be influenced by certain clerics who do not belong to the Society. 2. I am fully in agreement with the judgment that Your Holiness gave on the Second Vatican Council, on 6 November 1978, at a meeting of the Sacred College: "that the Council must be understood in the light of the whole of holy Tradition, and on the basis of the unvarying Magisterium of Holy Mother Church. 3. As for the Novus Ordo Mass, despite the reservations, which must be shown in its respect, I have never affirmed that it is in itself invalid or heretical. I would be grateful to God and to Your Holiness if these clear declarations could hasten the free use of the traditional liturgy, and the recognition of the Society of St. Pius X by the Church, and likewise of all those who, subscribing to these declarations, have striven to save the Church by perpetuating its Tradition. I beg Your Holiness to accept my profound and filial respect in Christo et Maria. + Marcel Lefebvre

From: https://www.sspxasia.com/Docuмents/Archbishop-Lefebvre/Case_for_Defence.htm

Anyway, that is a distinct question. Let's even put aside Pope John Paul II and Pope Benedict XVI for a minute, and focus solely on Pope Paul VI and Pope Francis for now.

1. Was Pope Paul VI Universally Recognized by the Teaching Church in 1965? Rev. Fr. Francis Connell, editor of the American Ecclesiastical Review after Msgr. Fenton, and personal friend of his, says that he was. "The whole Church, teaching and believing, declares and believes this fact, and from this it follows that this fact is infallibly true." This was in December 1965. What follows from this? That the sede-vacantist explanation of the Second Vatican Council is necessarily incorrect. 

2. Is Pope Francis Universally Recognized by the Teaching Church in this year of Our Lord 2020? To answer that question, we have to first ask and answer, where is the Teaching Church today? The teaching Church, the Catechism of Pope St. Pius X, and most Theologians will say, continues in Bishops appointed by the Pope to Office. Thus, either there is no teaching Church, which means the Church has defected, or only the Bishops appointed by the last 6 Popes constitute the teaching Church. Then, all these Bishops, who number about 5000+, both daily while offering Mass, and also when they would have received their Papal Mandate for appointment/consecration, manifested and do manifest their recognition of the Popes as Popes. Even Cardinal Burke recently said, he recognizes the Pope, prays for him as Pope in the Mass, and that this is not something he does lightly, but only upon careful reflection. 

Pope Benedict XIV teaches us, "But however it may be with this disputed point of ecclesiastical learning, it suffices Us to be able to state that a commemoration of the supreme pontiff and prayers offered for him during the sacrifice of the Mass is considered, and really is, an affirmative indication which recognizes him as the head of the Church, the vicar of Christ, and the successor of blessed Peter, and is the profession of a mind and will which firmly espouses Catholic unity. This was rightly noticed by Christianus Lupus in his work on the Councils: “This commemoration is the chief and most glorious form of communion” (tome 4, p. 422, Brussels edition). This view is not merely approved by the authority of Ivo of Flaviniaca who writes: “Whosoever does not pronounce the name of the Apostolic one in the canon for whatever reason should realize that he is separated from the communion of the whole world” (Chronicle, p. 228); or by the authority of the famous Alcuin: “It is generally agreed that those who do not for any reason recall the memory of the Apostolic pontiff in the course of the sacred mysteries according to custom are, as the blessed Pelagius teaches, separated from the communion of the entire world” (de Divinis Officiis, bk. 1, chap. 12)." https://www.papalencyclicals.net/ben14/b14exquo.htm

Does the Episcopal College, at the current time, pray for and commemorate the Pope as Pope in the Mass? If yes, then this is a sign that he is truly Pope, because by it, the Church "recognizes him as the head of the Church, the vicar of Christ, and the successor of blessed Peter" and that Her recognition, as Fr. Dom Prosper Gueranger says, "is a proof that, from that moment at least, the occupant of the Apostolic See is as such invested by God himself.” (Abbot Guéranger, O.S.B., The Liturgical Year , Vol XII, pg. 188)" 

Archbishop Lefebvre probably wavered a little on the question. Most of the time, His Grace was firmly against SVism. At certain times, he was unsure, true; yet, at other times, +ABL himself clearly expressed the idea, for e.g. on March 9 1980:

"Archbishop Lefebvre's attitude was made very clear in a letter to the Sovereign Pontiff dated 8 March 1980. It reads as follows:

Stop distorting the truth.  It is well known that Archbishop Lefebvre was most against sedevacantism during the early years of the Wojtyla pontificate.  After his bitter experience with Paul VI, he had hope.  But after further disillusionment and around the time of Assisi, he came very close to sedevacantism.

As Sean points out, Archbishop Lefebvre completely disagreed with the assertion that the legitimacy of these men is dogmatic fact.  Otherwise, even to entertain the possibility that the sedevacantists might be right would in fact be heretical.

Your continued assertion that this is dogmatic fact is tantamount to accusing Archbishop Lefebvre of heresy.  But, then, you live in a constant state of contradicition, picking and choosing things that back up what you want to believe.

You, XavierSem, have demonstrated yourself to be a schismatic.  You haven't provided a single reason why you cannot in good conscience join up with a group like FSSP who is in full communion with Rome, so you have no justification for the separation and are, therefore, a schismatic.  Did you get tossed out of the FSSP seminary or something?

Yeti, do you believe what the Catechism of Pope St. Pius X teaches: "43 Q. Of whom is the Teaching Church composed?
A. The Teaching Church is composed of all the Bishops, with the Roman Pontiff at their head, be they dispersed throughout the world or assembled together in Council ... 46 Q. Are we obliged to hear the Teaching Church?
A. Yes, without doubt we are obliged under pain of eternal damnation to hear the Teaching Church; for Jesus Christ has said to the Pastors of His Church, in the persons of the Apostles: "He who hears you, hears Me, and he who despises you, despises Me."https://www.ewtn.com/catholicism/library/catechism-of-st-pius-x-1286

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Of course I accept them. Do you grasp that what we dispute is not whether Catholics are obliged to obey the magisterium, but rather we dispute the identity of who composes the magisterium?
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By saying they are all heretics, you deny the Indefectibility of the Teaching Church. But then, you already deny it by Ecclesia-Vacantism.

The First Vatican Council saying there will be Shepherds and Teachers in the Church until the end of time is a refutation of both claims.

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I'm not convinced your interpretation of Vatican I is correct. Yes, theologians did teach that there would always be bishops in the Church, because they did not foresee the crisis we have today, but Vatican I was defining against the protestant heresy that denies the Catholic notion of the magisterium. They were saying St. Peter and the Apostles had no successors in their respective offices, so that there is no magisterium. Vatican I was defining that their offices were passed down and will continue to the end of time. It was speaking about *offices* of the papacy and jurisdictional bishops, not directly about the men holding those offices. Does this mean it didn't define that there would always be someone holding the office of bishop somewhere in the world? Not necessarily, but the burden of proof would be on you to prove this interpretation, since you're applying the words to a question it wasn't speaking to. And I certainly believe the offices of the pope and bishops will last until the end of time, so it's not at all clear to me that I am in violation of Vatican I.
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Cardinal Burke and Bishop Athanasius Schneider come to mind first; but then there is Cardinal Sarah, Cardinal Ranjith, Cardinal Ejik, and so many others.

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Now we're talking! Let's see. First of all, Bp. Schneider is an auxiliary bishop. He doesn't even claim any kind of jurisdiction, so he wouldn't be on the list. Regarding the others, how many of them accept this canon of the Council of Trent (Denzinger 956):
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Can. 9. If anyone says that the rite of the Roman Church, according to which a part of the canon and the words of consecration are pronounced in a low tone, is to be condemned, or that the Mass ought to be celebrated in the vernacular only, or that water should not be mixed with the wine that is to be offered in the chalice because it is contrary to the institution of Christ: let him be anathema.
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How many of them condemn ecuмenism? How many teach the traditional teaching that it is a grave sin to participate in the rites of non-Catholics? How many of them have condemned the phony marriage annulments that are given out by the thousands? How many of them teach that all nations have a grave obligation to enthrone Christ and His Catholic Church as the only true religion in their countries? How many of them denounced the canonization of Paul VI?
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For the record, I don't positively maintain that there are no bishops or cardinals in the Novus Ordo Church that don't have the Catholic Faith. I'm just saying that I don't know of any who do, and I think we would know if there were. There was one bishop who stood up to the heresies right away, and he has gone down in history as one of the most well-known bishops in the Church of the 20th century. Obviously I'm speaking of Archbishop Lefebvre. There isn't anyone like that today.
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EDIT: formatting