First, let’s explain what the Catholic Church means by infallibility: Infallibility is a protection that prevents those who are so protected from infallibly defining any false teaching on a matter of doctrine or morals. It is not a protection given to
all Church leaders. It is given to the pope specifically, and to the bishops as a collective whole when they teach in communion with the pope. The pope does not teach infallibly in every utterance he makes; he only teaches infallibly under certain limited conditions. Individual bishops, speaking on their own, do not exercise infallibility. For more information on the Church’s teaching about infallibility, please see the tract linked below.
Moving on, let’s take the more general question of why God might have given the Church the gift of infallibility. I submit that it may have been precisely in order to protect Church teaching on matters of doctrine and morals from the danger of its being explicated incorrectly by sinful human beings. In other words, infallibility was a protection given precisely in order to protect the Church from the effects of human corruption. Seen this way, infallibility is not a power tool given to certain sinful human beings to allow them to impose their will on the helpless faithful. Rather, it is a protection
for the faithful and
for Christ’s bride, the Church, that limits the ability of sinful leaders to manipulate Church doctrine to suit their sinfulness.
Now let’s look at the idea that the Christian faithful are unable to question their leaders when their leaders are living corrupt lives. If you look at Catholic history, you see that this is simply not the case.
St. Catherine of Siena, as only one example, powerfully rebuked the corruption of her day, acting as a moral conscience to the Church leaders with whom she interacted. Catholics are not forbidden from making known their concerns to their pastors, especially when certain leaders in the Church are not acting in accord with Christian morality. But there is a difference between rebuking a person who is acting badly (as Paul did to Peter,
cf.
Gal. 2:11-14) and denying that a properly-constituted authority has the ability to teach infallibly (which Paul never did).
As another example, which is a foreshadow of the infallibility entrusted to Christ’s Church, look at the story of the high priest Caiaphas, who was involved in condemning Jesus to death. He behaved in a corrupt manner by plotting to execute an innocent man; and yet, in his capacity as high priest, John records that Caiaphas made a prophecy of the atoning value of Christ’s death:
So the chief priests and the Pharisees gathered the council, and said, “What are we to do? For this man performs many signs. If we let him go on thus, every one will believe in him, and the Romans will come and destroy both our holy place and our nation.” But one of them, Caiaphas, who was high priest that year, said to them, “You know nothing at all; you do not understand that it is expedient for you that one man should die for the people, and that the whole nation should not perish.” He did not say this of his own accord, but being high priest that year he prophesied that Jesus should die for the nation, and not for the nation only, but to gather into one the children of God who are scattered abroad (
John 11:47-52).
Recommended reading:
Papal Infallibility
A Crisis of Saints by Fr. Roger Landry
Problems in the Church by Jimmy Akin