This is not proven true and one cannot appeal to the current claimants because it can't be proven that they were legitimately elected and therefore valid Popes to begin with.

it can't be proven that they were legitimately elected and therefore valid Popes to begin with.

This proves you don't quite understand what you're arguing against. A true Pope cannot OFFICIALLY teach error to the entire Church in an infallible capacity. Since we know that Vatican II taught error, infallibly in appearances since it is considered an Ecuмenical Council, we know that these men could not have been true Popes. Since error cannot be the official teaching of the Church, infallibly approved by the Pope, we have to assume they were not elected in the first place, thereby making them false claimants.
The fact that Vatican II did what it did, forces Catholics to look into the writings, statements, and actions of these men, prior to their election. Since we know that manifest heretics are not Catholics and therefore cannot be elected to the Papacy, we know they could not have been Popes. This resolves the issue of the appearance of DEFECTION of the Church, which is what it would have done were John XXIII and Paul VI validly elected Popes.

That an ecuмenical council which satisfies the conditions above (papal convocation and approbation + assembly of bishops) stated is an organ of infallibility will not be denied by anyone who admits that the Church is endowed with infallible doctrinal authority. How, if not through such an organ, could infallible authority effectively express itself, unless indeed through the pope? If Christ promised to be present with even two or three of His disciples gathered together in His name (Matthew 18:20), [color=rgba(0, 0, 0, 0.870588235294118)]a fortiori[/color] He will be present efficaciously in a representative assembly of His authorized teachers; and the Paraclete whom He promised will be present, so that whatever the council defines may be prefaced with the Apostolic formula, "it has seemed good to the Holy Ghost and to us."

A true Pope cannot OFFICIALLY teach GRAVE AND SUBSTANTIAL error to the entire Church in an infallible capacity.

It might be definitely understood, however, that the Catholic’s duty to accept the teachings conveyed in the encyclicals even when the Holy Father does not propose such teachings as a part of his infallible magisterium is not based merely upon the dicta of the theologians. The authority which imposes this obligation is that of the Roman Pontiff himself. To the Holy Father’s responsibility of caring for the sheep of Christ’s fold, there corresponds, on the part of the Church’s membership, the basic obligation of following his directions, in doctrinal as well as disciplinary matters. In this field, God has given the Holy Father a kind of infallibility distinct from the charism of doctrinal infallibility in the strict sense. He has so constructed and ordered the Church that those who follow the directives given to the entire kingdom of God on earth will never be brought into the position of ruining themselves spiritually through this obedience. Our Lord dwells within His Church in such a way that those who obey disciplinary and doctrinal directives of this society can never find themselves displeasing God through their adherence to the teachings and the commands given to the universal Church militant. Hence there can be no valid reason to discountenance even the non-infallible teaching authority of Christ’s vicar on earth.

It is, of course, possible that the Church might come to modify its stand on some detail of teaching presented as non-infallible matter in a papal encyclical. The nature of the auctoritas providentiae doctrinalis within the Church is such, however, that this fallibility extends to questions of relatively minute detail or of particular application. The body of doctrine on the rights and duties of labor, on the Church and State, or on any other subject treated extensively in a series of papal letters directed to and normative for the entire Church militant could not be radically or completely erroneous. The infallible security Christ wills that His disciples should enjoy within His Church is utterly incompatible with such a possibility.

Unless you are saying that you believe that it would be possible for a Pope to make a small error in an INFALLIBLE capacity, so long as it is not grave and substantial. Please clarify. Did you not read the end part of my sentence or are you saying what I just wrote (I hope it's not what I just wrote)?