Um, no. Ever heard of the Ordinary Universal Magisterium? The Extraordinary Magisterium (ie. ex cathedra statements, etc) is only one way that the Church is infallible.
And no again to the BOD and BOB argument. Those have nothing to do with false religions. It is NOT the same thing as saying that a false religion is a means of salvation.
Looks like I’m even more cynical than you! The so-called Ordinary Universal Magisterium is pretty much a product of Vatican II and its claim to infallibility wasn’t proclaimed infallibly then and certainly, in view of its late appearance, hardly carries the weight of such erstwhile traditions as the Immaculate Conception and the Assumption (which, of course, were later proclaimed ex cathedra).
On the false-religion comment, I tend to agree with you if the statement is in true context. Years of trying to see God’s will or just plain good in any part of VatII have forced me to try to justify what came of it in pre VatII outlooks and arguments. It’s a survival thing, I guess.