Strictly speaking, the Church does not, in fact cannot, "fail, falter and recover." If you mean popes or people or movements or theological theories or doctrines or priests, etc, yes of course they may change or fail. But there is no distinction between Christ and His Church in terms of moral infallibility. The Chruch's dogma cannot be wrong, as promised by Christ.
Catholism is a living religion, not a religion based on a book, but the Living Word, present in the world as Christ in his Church and through the Eucharist. The revealed truths of God (in His Church) have never altered.
The Church is a body of human beings, and it has all the weaknesses of human beings. Our Lord took upon himself that weakness. He failed to win over hiusown people. He came unto his own and his own knew him not. But by "sticking to his mission," he triumphed over the greatest failure of all, which is death. The Church is like Israel. He will never forsake themor us, but even when they had the ark of the covenant with them, they strayed. We likewise stray, but all have the assurance of victory if we follow the Way. It is not the Church that is the cloud by day and the pillar of fire at night, It is the Eucharist.
*I think you typed that without sufficient consideration