"If a man in leadership falls into sin, admits it, repents and turns from it, should he ever lead again in the same role?"
All men, save Jesus, are sinners. All men, save Jesus, are called to repent and turn from their sins. And only men are called to lead in the church. As such, if we are going to have leaders, that is, elders, and deacons in the church, we had better leave room for repentant elders and deacons. The only thing worse and the only other thing possible is unrepentant elders and deacons.
That said, I suspect the question, while vague, is aiming at something a bit more particular. What do we do with a pastor who has committed adultery? What do we do with a deacon who has embezzled the church’s funds? If they repent, it would seem we are called to forgive. And doesn’t forgiveness mean we act as though it never happened?
Yes, of course we are to forgive the repentant. That doesn’t mean, however, that we are to act as though it never happened. When we forgive we do not forget as if we had amnesia, or as if there is nothing to be concerned about. Instead we forget in the sense that we no longer hold the sin against the sinner. We do not hold a grudge against them. We love the repentant. We embrace the repentant, and we seek to help not just the repentant, but those whom they have wronged. We do not require the embezzler to wear a scarlet E for the rest of his life. But we do not either leave him alone to count the offering. We would be poor stewards of his soul and the kingdom’s funds were we to leave him to his temptation.
Consider how God’s law deals with adultery and divorce. Were I unfaithful to my wife, and were I to repent for such a sin, she would have an obligation to forgive me. She would not, however, have an obligation to stay married to me. Adultery is biblical grounds for a divorce precisely because it is such a betrayal of a trust that future trust is hard to come by. The victim is to forgive. The adulterer is forgiven, But the divorce can still happen, and is still laid at the feet of the adulterer. He is the one who broke the covenant. The victim is free to acknowledge that reality by seeking the legal divorce.
One could argue, and indeed some have, that a pastor who is guilty of infidelity is to be forgiven, but as with marriage itself, has so betrayed the trust inherent in his office that it would preclude his future service as a minister of the gospel. Others, perhaps pointing to Peter’s betrayal of Jesus, and Jesus’ admonition after his repentance that he strengthen the brethren, that a pastor in such a circumstance is oddly even more empowered to serve as a minister of the gospel, having experienced its power so immediately. The danger is, in both positions, papering over our emotional response with pious words. That is, too often the pastor is put out not because it is the right thing, but because of anger, because we haven’t honestly forgiven. Even more common we are fearful of how the church would fare without our pastor, and so keep him on, even cover up for him, and excuse our fear by baptizing it in “forgiveness” and “grace.” Because we are all sinners our temptation is always to do what we want to do. Because we profess Christ, we then cover our desires with rationalizations.
God is good. God can and does not only forgive us, but can and does cleanse us from all unrighteousness (I John 1:9). That said, a man who has proven his willingness to betray his family is more likely than one who has remained faithful to walk into adultery again. A man who has betrayed his office sexually, is likewise more likely to do so again. My counsel would be to remove the man from office. But it is just that, counsel. I cannot claim that the Bible commands it, nor that if forbids leaving such a man in office should he repent.