The fact of the matter is that it is a matter of faith that there will be successors of the apostles until the end of time, so the death of the bishops ordained and appointed during Pius XII's reign means nothing other than they are no longer successors of the apostles-- at which point we would know with certainty of faith that there are successors:
-John XXIII (as pope or through supplied jurisdiction of an antipope)
-Through Paul VI (most likely supplied jurisdiction for an anti pope, esp. after 1965)
-Theoretically (and very theoretically) successors from JPII, BXVI and possibly (though even more theoretically) through Francis
-Eastern Rite Bishops (via jurisdiction supplied to an antipope or not)
-Very, very, very theoretically to traditional bishops-- I would not make this claim but I know that some do
-Secret hierarchy via Red Curtain bishops, Siri, etc.
A teaching which is of the faith cannot be denied. There are successors to the apostles; we may not know who they are but that hardly means they do not exist. The Church began in an upper room, it is perfectly conceivably that she will "end" in a lower room, in the catacombs as it were.
Good post. I would say though that with traditional bishops that any claim to them being formal successors is an error.
All of the traditional bishops know this, which is why they have always denied that they have jurisdiction and formal apostolic succession. The only bishop that ever claimed jurisdiction was the late Bp. Vezelis.
The only proponents of this view are certain laymen, who, in my opinion, do not understand the relevant theology of Apostolicity.