Let us suppose for a moment that the Sedevacantists are right. The chair of Peter has, by some form of apostasy or other, been vacant for a significant time now. In fact, the exact date and cause doesn't even matter in this line of reasoning. Even if you just suppose that the current claimant to the throne of Peter, one Jorge Bergoglio, has no such rightful claim in any form, actual or potential, the foregoing argument still holds on a carry-forward basis.
Now, if this is true, what are our actual obligations as faithful Catholics? Why, we are to find the highest remaining authority in the Church that has not gone along with the apostasy that led to this vacancy in the seat of Peter and follow that man. We have not always been promised a pope. Our Lady warned us when she appeared at La Salette that "Rome will lose faith and become the seat of the Antichrist." We do not have popes during the time between the death of one and the election of another; this is the origin of the term sedevacante. While it is an obligation for us to hold that the supreme earthly authority and rightful exercise of power in the Church belongs to the successor to Saint Peter, in absence of such a one, the hierarchical structure of the Church, by its very nature, does not break down. There may be limits on what can be done in absence of a pope, but God's authority still remains in the Church, who still possesses the keys which our Lord gave to Saint Peter and against which our Lord promised the gates of Hell shall never prevail.
Viewed in this light, it is somewhat silly the amount of ink that has been spilt by traditional Catholics over the issue of sedevacantism. It is disordered in light of the real issue, identified over and over by Our Lady, apostasy. And in any event, while the philosophical truth of the matter is indeed important, the practical outcome can easily remain the same. Even if you adopt the sedevacantist position, it would seem to make the most sense to follow the Society of Saint Pius X, as founded by the highest ranking Church authority that openly opposed the current apostasy who in keeping with the rules set by previous legitimate papal authority created an institution that unimpeachably carries the apostolic succession forward.
Briefly, let us spend a moment dismissing the claims that the Society, despite by all outward appearances keeping the faith, somehow at one time or another found itself infiltrated by a mason or two rendering its effort in all or part void. As Church scholars have argued, since there is no way for earthly creatures to ascertain the intention of another such creature, mere objective participation in the rites of the Church, as they have been lawfully codified and always been celebrated, is enough to assume the proper intention. This is clearly the case, or the enemies of the Church would have never endeavored to alter the rites of the Church. Satan, ever the most cunning of God's creatures, would have seen all the more fit to leave the old rites in place and merely infiltrate the Church with men who would withhold their intention in performance of the old rites. No! - no! - no!!! This was not an avenue for undermining the Church. There is something inherent, objectively true in the traditional rites of the mass that demonstrate integrity between action and thought. There are some actions you cannot perform without making an admission of their own truth. We know this even from simple, secular forms of study, such as economics. "Man acts," and to endeavor to disprove the truth of this statement would itself be an action, thereby proving its truth. It is impossible to participate in the traditional rites, without admitting their truth. This is not the same of the new rites of the mass, and the very reason the enemies of Christ saw fit to implement them.