The papacy is credited by conservative Catholics as the rock that has kept the Church steady and faithful in the winds of historical change and heresy. If it instead becomes an obvious, visible, instrument for the destruction of Catholic tradition, conservative Catholics will be forced to revise that assessment. They will either be forced (to some degree at least) into the position of Catholic traditionalists, who already view the modern papacy this way because of the changes of Vatican II, or they will flee to Eastern Orthodoxy, which views the modern papacy as the fruit of centuries of papal error.
While the former is a smaller bridge to cross, there's a certain absurdity to anti-papal traditionalism, that will make Orthodoxy seem more consistent (and stable and attractive) to many. I would expect especially that the waves of converts from Protestantism that were once attracted by the apparent permanence of Catholicism and the strength of the papacy, having found that their trust has been harshly betrayed, will move on to Orthodoxy. (And, for those Protestants who are now looking for a more historical form of Christianity, Orthodoxy is going to start to look more attractive and viable in comparison to Rome.)
One wonders too about the bishops with traditionalist leanings appointed by Benedict XVI. Will they hang in there, providing some cover for traditionalists? (Will some of them even defect to Orthodoxy, perhaps having an overly expansive notion of papal infallibility popped by reality?) Because the papacy has attained complete control over the appointment of bishops, Catholic traditionalism could find itself increasingly forced into institutional estrangement from the papacy and the creation of a counter-hierarchy, and conservative Catholic who formerly stayed within the mainstream Vatican II fold may find themselves fleeing to traditionalism just to take advantage of those institutions to avoid increasing and seemingly terminal liberalism within the mainstream Church. (Eastern Catholic Churches may also serve as a refuge, as they already for many Latins looking for formal liturgy and traditional teaching and practice.)
Maybe things won't get this dark, and there are enough good bishops to keep things from going off the rails too quickly, but if Francis gets to work promoting liberal bishops and stacking the college of cardinals, it's hard to see a way to correct the direction of the hierarchy from within.
