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Xavier, that Pius XII passage is making a distinction between 1) infallible teaching and 2) papal decisions (ie “passing judgement”).  He was saying that it is wrong to ignore papal judgments (ie legal/govt decisions) because they aren’t “infallible doctrine”.  He was also saying that some some encyclicals “pertain to doctrine” so we must give respect to them (which, up til that time in history, 1900 yrs, there has never been error in an encyclical).
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V2 didn’t pass judgement on anything (which is part of the pope’s governing power, not his teaching power).  It did “pertain to doctrine” but giving assent (temporarily) does not mean we accept error (long term).
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As +Vigano stated, we’ve tried to make V2 consistent with Orthodoxy and it doesn’t work.  So we cast it out into the darkness. This is consistent with what Pius XII wrote above. 

Anyway I thought it was dogma that there would be popes until the end of time. But the sedes argue that one away as well. Except for a hundred years here or there.

Is perpetually, in the context you allude to, broken every time a Pope dies?  Of course not!  Whose “time” matters most?  Ours or God’s?  I submit that would be God’s “time” is much more relevant than ours.  61 years and 9 months, 100 years, 500 years is hardly more to God than the blink of an eye—hardly more than the interim when a Pope passes.

This is the most wrongly-quoted Pius XII passage in history.  Both sedes and (anti-sede) Xavier quote it the same, but it's a lack of reading comprehension that produces a wrong conclusion.
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20. Nor must it be thought that what is expounded in Encyclical Letters does not of itself demand consent,

Of course, an encyclical demands assent.  But what kind?  Certainly not "certainty of faith" which is only for infallible/doctrines.  "Religious submission" maybe?  Which is defined as assent to superiors but which still allows questions/critiques if the encyclical is confusing.
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since in writing such Letters the Popes do not exercise the supreme power of their Teaching Authority.

Obvious.
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For these matters are taught with the ordinary teaching authority, of which it is true to say: "He who heareth you, heareth me";[3]

Ordinary teaching authority is neither infallible, nor error-free.  It is a reiteration of defined truth.  A re-teaching of dogma.  Mistakes can be made.  Even heresies.  But...Pius XII is saying that we USUALLY trust such encyclicals BECAUSE (see the next phrase below, which is ALWAYS forgotten)...
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and generally what is expounded and inculcated in Encyclical Letters already for other reasons appertains to Catholic doctrine.

Pius XII is saying that we trust encyclical letters because..."GENERALLY" (but not always.  Not an absolute.  Not 100%) these docuмents pertain to doctrine.  AND...one assumes that such doctrinal writings are ORTHODOX (because we've never before in history had an infiltration into the Church that we've had today.).
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So let's sum up the reasons why we trust encyclical letters (in normal orthodox times):
1.  Because we owe respect and 'religious submission' to our fallible pope.
2.  Because (normally) a pope has orthodoxy as his goal and is trying to teach the faithful good things.
3.  Because (normally) an encyclical is on doctrinal matters, so we give it the benefit of the doubt that it is orthodox, due to trust we have in #1 and #2 above.
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None of this applies to the V2 popes or their encyclicals. 
1) We give them 'religious submission' but we also have the right to be wary of their lack of orthodoxy and to critique their less-than-true writings. 
2) The V2 popes have proven that orthodoxy is not their ultimate goal, nor is re-teaching doctrine, but want to "pastorally apply doctrine" in catholic lives in "new and unique ways for the modern man". 
3) V2 encyclicals have not been "GENERALLY...PERTAINING TO CATHOLIC DOCTRINE", as they've been mixed with new-age ideals, political concerns, and pastoral directives.
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Thus, V2 encyclicals are of the lowest level of the magisterium, which is to say they are 100% fallible and capable of heresy because when you mix doctrine with non-doctrine (politics, pastoral directives, humanism/socialism) then such writings are no longer doctrinally-focused but only opinion, theory and agenda.