Bp Williamson's letters make it quite clear that he understands we are dealing with TWO bodies -- and one of them is NOT Catholic (although it has yet to come under the knife, i.e. be juridically severed).
As I've written before, this theology of the Two Churches bugs me. It comes from the thinking of Archbishop Lefebvre, to whom Bishop Williamson of all the current SSPX bishops has the greatest devotion.
Certainly the NO establishment has a material continuity with the Catholic Church, and perhaps that's what they're getting at without actually saying so. (cf. Bishop DesLauriers' material/formal distinction). But the Orthodox have similar material continuity with certain Apostolic Sees. Yet it doesn't make than any more Catholic.
If this explanation for the crisis is to hold any water, they have to at least come up with some kind of theological explanation for it. If pressed, I suspect they'd have to come up with a formal/material distinction similar to the Thesis Cassiciacuм.
Eternal Rome = formal Church
modernist Rome = material / juridical Church
But the SSPX needs to elaborate on this theologically rather than speaking in mystical or poetic language.
Archbishop Thuc also hinted at this in his famous declaration of
sede vacante.
Such a distinction could actually be used as a rational justification for the approach they take. Yet to elaborate this would lead to a formal sedevacantism, elsewhere called sedeprivationism, so they won't go that far and thus get mired in some really problematic positions, such as that the Holy Catholic Church could promulgate a harmful, destructive, "bastard" rite of Mass which undermines the faith and cannot be attended in good conscience.
I suspect that in using such language most SSPX folks are implicit sedeprivationists.