This is one the many teachings that is debated within Rome as to whether they were infallible. Another question is, how evident was it that those closest to the statements understood it to require formal submission to Rome?
All told, the sad fact, which is testified to both doctrinally and empirically, is that there is seldom found salvation within Rome, which effectually promotes faith in one’s merits (”by those very works ..to have truly merited eternal life” - Trent )and the power of the church for salvation, rather than contrite repentant faith in Christ and His sinless shed blood, which while multitudes are manifestly born again by believing.
And it the church that works to convict souls of their utter inability to either escape Hell or gain Heaven by any dependence upon the works or church, but upon the Biblical Christ, that is indicative of a true church. And whose authenticity does not rest upon formal ecclesiastical linkage, any more than that of a true Jew rests upon physical genealogy to Abraham. Rather in both cases it rests upon Abrahamic type faith in the apostolic gospel of the grace of God.
And since it is upon faith in Christ that one is baptized into the church (1 Cor. 12:13), there is no one saved outside of it.
“All told, the sad fact, which is testified to both doctrinally and empirically, is that there is seldom found salvation within Rome”
Of course you use the word Rome to represent your own distorted perception.
“...and the power of the church for salvation, rather than contrite repentant faith in Christ and His sinless shed blood.”
How can you have contrite repentant faith in Christ and not attempt to live by His Laws?
“All told, the sad fact, which is testified to both doctrinally and empirically, is that there is seldom found salvation within Rome”
It sounds more like an impersonation of God, actually a usurpation of God’s power of judgement. It might make a great quip “I’m not God but I play Him on the internet.”