They will find themselves questioning how the Church that claims to be the Church instituted by Christ could have such abysmal leadership. One would think that Christ would protect his own if they truly are his own. The atheists argue exactly the same way against the very existence of God. Just look at all the suffering children around the world... IF they are all creatures of God, why doesn't He care? The same answer applies here. God's providence is at work. Read St. Augustine's City of God book XVIII, especially the later chapters, where Augustine describes how Satan raises up heretics and wretches in the Church to destroy the faith of the Church, yet God uses the same heretics and wretches to build up virtue in the saints. All things work for the good of those who love Him... if we respond with confidence in the power of the Holy Spirit. The signs of the Spirit's action are everywhere. We only see the effects of the Spirit. The Saints toughed it out alot more than we have. St. John of the Cross was imprisoned by his own religious order for over a year. It was in prison that he went throught he dark night of the soul, and then the Carmelites were truly renewed.
By the way, where in Scripture does it ever say that when the going gets tough, one should just leave the Church. I see prescriptions to rebuke, exhort, even excommunicate, but it nowhere says "just leave..." That is an unbiblical practice the Prots have going for themselves.