As for a sanatio in radice of an illegitimate election, I do not buy it.
In that case, you disagree with the doctrine of Universal Acceptance, as explained by Cardinal Billot. Pope John XXIII's (and then Pope Paul VI's) Universal Acceptance is a sufficient proof that Cardinal Siri did not remain Pope, even if he was elected Pope to begin with. Cardinal Siri must, then, have resigned.Fr. Connell, AER, 1965: "The whole Church, teaching and believing, declares and believes this fact, and from this it follows that this fact is infallibly true. We accept it with ecclesiastical – not divine – faith, based on the authority of the infallible Church" in response to the question, "What certainty have we that the reigning Pontiff is actually the primate of the universal Church – that is, that he became a member of the Church through valid baptism, and that he was validly elected Pope?"Fr. Boulet, SSPX, 2004: " the most important reason why we must discard the "Pope Siri" theory is the fundamental principle that a peaceful acceptance of a pope by the Universal Church is the infallible sign and effect of a valid election. All theologians agree on that point. Cardinal Billot says: "God may allow that a vacancy of the Apostolic See last for a while. He may also permit that some doubt be risen about the legitimacy of such or such election. However, God will never allow the whole Church to recognize as Pontiff someone who is not really and lawfully. Thus, as long as a pope is accepted by the Church, and united with her like the head is united to the body, one can no longer raise any doubt about a possible defective election"
Supposing someone says Pope Pius XII, or Pope Pius IX, or anyone else, never became Pope, because there was some other alleged secret Papal claimant somewhere, does their Papacy really become doubtful because of that? Most certainly not, otherwise it is goodbye to dogmas like the Immaculate Conception and the Assumption of the Mother of God. Papacies cannot be denied or doubted, as Cardinal Billot explains, once Universal Acceptance is verified.
As for cuм Ex, it is speaking about the Cardinals. That is plain from the text itself, and also from Lateran V, which decreed similarly on simony. As Cardinal Billot in the case of Pope Alexander VI (who was accused of both heresy and simony), universal adherence proved he was not a heretic, and any fault he or others committed was healed in the root at the moment of such acceptance. Universal acceptance can be either a cause of convalidation or an effect of a valid election. What is certain is that, after UA, no doubt can be raised. This explanation is also given by Rev. Father Dom Prosper Gueranger, “The inevitable play of human passions, interfering in the election of the Vicar of Christ, may perchance for a while render uncertain the transmission of spiritual power. But when it is proved that the Church, still holding, or once more put in possession of, her liberty, acknowledges in the person of a certain Pope, until then doubtful, the true Sovereign Pontiff,
this her very recognition 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)"
Re: Pope St. Martin I. I agree with the CE's explanation: "
What happened in Rome after his departure is not well known. For a time the Church was governed in the manner usual in those days during a vacancy of the Holy See, or during the absence of its occupant, viz., by the archpriest, the archdeacon, and the primicerius of the notaries. But after about a year and two months a successor was given to Martin in the person of Eugene (10 Aug., 654).". Anyway, a doctrine is not to be revised based on a doubtful historical point. Only one thing could support Bene-Vacantism, or Siri-Vacantism, or such similar theories; if the person who is claimed to still be Pope, maintained his claim to remaining Pope and not having resigned, refused to recognize the opposing claimant, and held a certain section of the Church in obedience. This was the case, for e.g. during the Great Western Schism. And once Pope Martin V received UA, the GWS came to an end.