I can't reply to everything that M. Davies wrote in this article right now, but I must make a few observations:
I suppose his salient point is that while sede vacante is not intrinsically impossible, he believes one of this length (so, say forty years or so) is impossible because "there would be no visible source of authority capable of convoking a conclave to elect a new pope." Just because Davies doesn't know who the hierarchy are doesn't mean they don't exist. It's a matter of fact that bishops still exist who were lawfully appointed, and it's a matter of faith that the hierarchy cannot cease to exist: this would be a defection of the Church. If one considers the Church at Pentecost, the hierarchy was unknown to the VAST majority of the world. Was it invisible? Or, consider the Japanese Catholics who were cut off from any clergy for what, a hundred years? Two hundred years? The hierarchy did not cease to exist. Nor did it while the Maronites were cut off from it. I have to admit that of all the apparent difficulties posed by an indefinite interregnum, this is the one to pick on, but once one correctly understands that "visible" doesn't mean the same as "known to you, specifically" then the difficulty disappears. For anyone who accepts that the revolutionary victory of the modernist mafia at VII and in the wake of it was the result of ʝʊdɛօ-Masonic scheming and conspiracy, likely a plot that at least vaguely existed for an hundred years or longer, it should not be difficult at all to accept that whatever members of the hierarchy remain are drastically diminished in the public realm, known to very few-- for with them lies the continuation of the Church as a body, and the enemies of the Church want nothing more than to set their heads on pikes.
Davies says "the Church that [Jesus Christ] founded cannot fail, for it is indefectible (i.e. it cannot fail). It will continue to exist until the Second Coming as a visible, hierarchically governed body, teaching the truth and sanctifying its members with indubitably valid sacraments." Of course this is true. But would he have us believe that this Church is the Novus Ordo? Don't make me laugh. Teaching the truth, sanctifying its members with indubitably valid sacraments? The Novus Ordo doesn't even fulfill ONE of these requirements, much less all three! Teaching the truth? Unless VII is the truth, strike one. Sanctifying its members? Unless the NOM is pleasing to God as the unbloody sacrifice of Calvary as a propitiation for our sins, strike two. Indubitably valid sacraments? The changes to holy orders cast doubt onto the validity of all sacraments requiring a sacramentally imbued minister (confession, the mass, holy orders and extreme unction), strike three-- the Novus Ordo Church is out!
It is true that there is disagreement on the part of theologians as to whether or not a pope could fall into heresy, but none of the theologians who entertain the idea say that if he did, he would retain office. And the idea that a pope can't become an heretic (and therefore lose his office) is certainly nothing more than a theological opinion, unlike the effect of heresy (severance from the Body of Christ, rendering one a non-Catholic) which is an effect of divine law. So it is wrong for Davies to preface "if a pope became an heretic" with "per impossible." It's not impossible. There are just a few theologians who don't think it could happen-- but not thinking it can happen is not the same as thinking it's impossible! If St Robert Bellarmine thought it was impossible and violated the deposit of faith, he wouldn't entertain what would happen if a pope DID become an heretic!
Then Davies conflates "deposing" a pope with simply recognizing he is an heretic, and therefore not the pope. If a person looks at Rome, and sees that the man who purports to be pope is not Catholic, and therefore can't be pope, he does not "depose" the pope. He realizes a fact that is apparent, a fact that causes moral certainty for him on the state of the Holy See, and responds in kind. Davies would have us believe that some sort of declaration is necessary to arrive at this conclusion, but this is not the case. Canon 188/4 teaches that an office holder who publicly adheres to a false religion resigns his office. The same CIC teaches that by the fact of their heresy, public heretics incur a latae sententiae excommunication (this effect has the same result-- loss of office-- but is a distinct effect from canon 188). Thirdly, by Divine Law (both scripture and the universal ordinary magisterium) heretics are not Catholics and non-Catholics cannot participate in the economy of salvation, which includes the governance of the Church (again, this effect is distinct from the prior two). Finally, public heretics may have an excommunication inflicted upon them by a lawful superior. These are four distinct effects of an office holder who is an heretic. Davies is only addressing the last of these four. He briefly makes an attempt at distinguishing a formal heretic (otherwise called manifest, public, or simply heretic) and one who materially holds an heresy. The distinghisghing mark between these two very different species is pertinacity, which is knowing that the Church teaches X, and electing to teach or profess Y (which is contrary to X) publicly. It is preposterous to think that any of the conciliar pontiffs don't KNOW that the Church teaches there is no salvation outside of Her, or that there is ONE Church of Christ, the Catholic Church. Because they have all publicly denied this and professed something contrary to both of these doctrines.
Davies plays a copy and paste game, conveniently leaving something important out. He quotes Bouscaren and Ellis "...by divine and Catholic faith must be believed all that is contained in the written word of God or in tradition, that is, the one deposit of faith entrusted to the Church and proposed as divinely revealed either by the solemn Magisterium of the Church or by its Ordinary Universal Magisterium." In the next sentence, Davies says "No doctrine is to be considered as dogmatically defined unless this is evidently proven," as if to make the reader conclude that nothing that hasn't been solemnly defined is to be regarded with divine and Catholic faith. HOWEVER: he is not citing the whole passage. I own Bouscaren and Ellis' commentary. The sentence that goes between these two is "To pronounce a solemn definition is the part of an ecuмenical council or of he roman pontiff speaking ex cathedra." THAT is what must be proved to be "dogmatically defined." The ordinary magisterium must still be believed by divine and Catholic faith, even though much of it is NOT dogmatically defined in this way. Additionally, I have no idea what he means in his (13) footnote because p 724 deals with forbidden books.
As regards the doctrine of necessity, it treats epikeia and how according to circuмstance, a given law may cease to bind; taking into consideration the benevolence of the lawmaker, and that in a given situation the lawmaker would not wish the law to bind, a decision is made contrary to the letter of the law while maintaining its spirit. The docuмent put out by the SSPX is a good synopsis of this doctrine, but it does not at all address whether or not an heretic (who is not Catholic) can hold office in the Church.