I said that there have always been successors. I didn't say anything about the periods between successors, which we know has occurred. But the periods between successors didn't occur because laymen took it upon themselves to proclaim that there was or is no pope. You are aware of that, I hope.
In both scearios, R&R or SV (or in between with sedeimpoundism and sedeprivationism), you have "laymen [taking] it upon themselves" to reject 60+ years of alleged Papal Magisterium, an Ecuмenical Council, the Church's Rite of Public Worship, and the canonizations of saints. So you can't play the old "laymen [taking] it upon themselves" game.
So the only question here is between the explanations for this situation. Unfortunately, the (classic) R&R position destroys the very foundations of Catholicism, that the Church, the Magisterium, the Mass, the Sacraments, and canonizations cannot go off the rails.
This dispute between R&R and SV is actually nicely summed up in the teaching of Vatican I regarding the papacy. Vatican I was (as all Ecuмenical Councils) inspired by the Holy Spirit, and in this case timed precisely to prepare us theologicaly for what was to come in the Vatican II era.
R&R (as per usual) cite the "perpetual succession" teaching in Vatican I, but the completely ignore the following:
So the fathers of the fourth Council of Constantinople, following the footsteps of their predecessors, published this solemn profession of faith: "The first condition of salvation is to maintain the rule of the true faith. And since that saying of our lord Jesus Christ, You are Peter, and upon this rock I will build my Church, cannot fail of its effect, the words spoken are confirmed by their consequences. For in the Apostolic See the Catholic religion has always been preserved unblemished, and sacred doctrine been held in honor.
...
And since before all others [the Holy See] has the duty of defending the truth of the faith, so if any questions arise concerning the faith, it is by her judgment that they must be settled.
It was for this reason that the bishops of the whole world, sometimes individually, sometimes gathered in synods, according to the long established custom of the Churches and the pattern of ancient usage referred to this Apostolic See those dangers especially which arose in matters concerning the faith. This was to ensure that any damage suffered by the faith should be repaired in that place above all where the faith can know no failing.
Indeed, their apostolic teaching was embraced by all the venerable fathers and reverenced and followed by all the holy orthodox doctors, for they knew very well that this See of St. Peter always remains unblemished by any error, in accordance with the divine promise of our Lord and Savior to the prince of his disciples: "I have prayed for you that your faith may not fail; and when you have turned again, strengthen your brethren.
This gift of truth and never-failing faith was therefore divinely conferred on Peter and his successors in this See ...
So you selectively cite the one phrase about "perpetual successors" but have zero problem throwing all of the above into the trashbin.
Those of you who do this have become Old Catholics. In fact, those who reject SVism, like the heretic John Pontrello, often end up in Orthodoxy, rejecting Vatican I explicitly.
So either the Magisterium, the Mass, the Canon Law, the cult of the saints, can all become so thoroughly corrupted that Catholics are justified, nay, rather, required in conscience, to sever communion with the Holy See ... or else the Holy See can remain vacant for longer than you claim.
We have the famous passage from the Jesuit theologian Father Edmund O'Reilly, who stated that it's not incompatible with the promises of Christ to have an extremely lengthy vacancy of the Holy See. Meanwhile, claiming that the Magisterium, the Mass, etc. can go corrupt ... those are absolutely incompatible with the promises of Christ regarding the Papacy and the Magisterium and the Church.
Of course, if you accept a variation on either sedeprivationism or sedeimpoundism, you can retain both the material continuity of the papacy AND the dogmatic teaching of Vatican I regarding the unfailing faith of the Holy See.
But you so pertinaciously adhere to your Old Catholicism, that you might as well just admit it and go find an Old Catholic chapel. There's no difference between your brand of R&R and Old Catholicism.
Thus, please prayerfully consider the position articulated by Father Francois Chazal ... lest you scuttle your faith while petending to be defending it.