“...a Church that...”
You know, I presume, that the Church, being the spotless Bride of Christ, can not and does not teach error.
Wayward priests and prelates, by contrast, can and often do.
Christ’s Church exists, always exists, is always both visible and discernable, and of course the gates of hell will never prevail against her.
Does SSPV claim it is the Church?
Although they profess to carry on the traditional pre-Vatican 2 faith, I don't think they claim to be the Catholic Church exclusively.