The man elected is instantly the true pope, yes, but just because the news tells you an election took place doesn't mean that election was actually valid. Anti-pope Clement VII was elected by real cardinals with much pomp and circuмstance, and half of Europe would've told you that he was the validly elected true pope. Was he? Clearly not. You need to determine if an election was actually valid, which is not something your quote will help you with because that's not what it's even about. You just throw that quote out in every single thread without a single clue as to what it's actually referring to, and then act as if "the pope is the pope" is actually a useful or meaningful statement at all.

The full quote, including the part that you ALWAYS cut out because it doesn't suit your agenda is:
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After this agreement has been furnished within a time limit to be determined by the prudent judgment of the Cardinals by a majority of votes (to the extent it is necessary), the man elected is instantly the true Pope, and he acquires and can exercise full and absolute jurisdiction over the whole world . Hence, if anyone dares to challenge the docuмents prepared in regard to any business whatsoever that comes from the Roman Pontiff before the coronation, We bind him with the censure of excommunication to be incurred ipso facto

Here's how I answer the question posted by the OP.

I pretend that I'm living during the reign of Pope Pius XII, say, in 1950.  Is there a Catholic anywhere in the world who thinks that Pius XII is not Pope?  There's no dispute, no disagreement, no doubt, just peace of mind and calm regarding this identity.

Is this really what we have with Jorge Bergoglio ... when some even in the Novus Ordo are wondering if he's Catholic?

To play devil's advocate, the Old Catholics would assert there certainly was dispute and disagreement.

Well, that was a different era.  Which is why I used Pius XII.  Now, I'm not sure the Old Catholics ever questioned that he was the Pope.  They just didn't accept his teaching.  I'm not sure that's the same thing.

Now, let's say that the Old Catholics had gone all sedevacante, and argued that Pius IX was no longer pope due to being guilty of heresy on the infallibility question.  Their opinion would not matter, since they would have been heretics.  That's similar to how I feel about the vast majority of the Novus Ordo hierarchy who refuse to reject Bergoglio as a non-Catholic.  Why should their opinion be taken into any more account than that of, say, these hypothetical sedevacantist Old Catholics?

Well, those are interesting case studies, Clemens.

Now, I am in fact a dogmatic facter, in the sense that it's logically (theologically) certain that the legitimacy of a pope must be dogmatically certain, or else nothing they define as dogma can be dogmatically certain.

What's at issue is what criterion determines whether the legitimacy of any given pope is dogmatic fact.  It has always been held to be universal peaceful acceptance, but I will read up on and consider the case studies you posted.

But even while upholding universal peaceful acceptance, I do not consider rejecting Bergoglio to be heresy, since he decidedly lacks such universal peaceful acceptance by Catholics.

Perhaps the answer is that John of St. Thomas did not correctly think through the quoad se vs. quoad nos distinction.

If a Pope were elected and universally accepted, then he's Pope by dogmatic fact.  When a subsequent man comes along, deposes him, puts him in jail, assumes the See with new "universal acceptance", the universal acceptance means nothing, for the quoad nos cannot override the quoad se.  No subsequent universal acceptance of another man can cause the deposition of the original pope, for that would be a flavor of Conciliarism, where the Church can effectively depose popes.